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Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems (2024)
Journal Article
Radley, T. J., Houston, P., & Hubbard, M. E. (2024). Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems. Mathematics in Engineering, 6(1), 192-220

In this article we consider the application of Euler’s homogeneous function theorem to- gether with Stokes’ theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two and three dimensio... Read More about Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems.

Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes (2022)
Journal Article
Congreve, S., & Houston, P. (2022). Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes. Advances in Computational Mathematics, 48(5), Article 54. https://doi.org/10.1007/s10444-022-09968-w

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated poly... Read More about Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes.

Linearization of the Travel Time Functional in Porous Media Flows (2022)
Journal Article
Rourke, C. J., Houston, P., Rourke, C., & van der Zee, K. G. (2022). Linearization of the Travel Time Functional in Porous Media Flows. SIAM Journal on Scientific Computing, 44(3), B531-B557. https://doi.org/10.1137/21M1451105

The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the postclosure safety assessment of deep geological storage facilitie... Read More about Linearization of the Travel Time Functional in Porous Media Flows.

Gibbs phenomena for Lq-best approximation in finite element spaces (2022)
Journal Article
Houston, P., Roggendorf, S., & Van Der Zee, K. G. (2022). Gibbs phenomena for Lq-best approximation in finite element spaces. ESAIM: Mathematical Modelling and Numerical Analysis, 56(1), 177-211. https://doi.org/10.1051/m2an/2021086

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as L q-type Sobolev spaces. For q → 1, these methods have demons... Read More about Gibbs phenomena for Lq-best approximation in finite element spaces.

High-Order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations (2021)
Book Chapter
Antonietti, P., Facciola, C., Houston, P., Mazzieri, I., Pennesi, G., & Verani, M. (2021). High-Order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations. In D. Di Pietro, L. Formaggia, & R. Masson (Eds.), Polyhedral methods in geosciences (159-225). Springer

An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids (2020)
Journal Article
Antonietti, P. F., Houston, P., Pennesi, G., & Suli, E. (2020). An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids. Mathematics of Computation, 89, 2047-2083 . https://doi.org/10.1090/mcom/3510

In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear system of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial diffe... Read More about An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids.

Two?Grid hp ?DGFEMs on Agglomerated Coarse Meshes (2019)
Journal Article
Congreve, S., & Houston, P. (2019). Two?Grid hp ?DGFEMs on Agglomerated Coarse Meshes. PAMM, 19(1), https://doi.org/10.1002/pamm.201900175

We generalise the a priori error analysis of two?grid hp?version discontinuous Galerkin finite element methods for strongly monotone second?order quasilinear elliptic partial differential equations to the case when coarse meshes consisting of general... Read More about Two?Grid hp ?DGFEMs on Agglomerated Coarse Meshes.

The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method (2019)
Journal Article
Houston, P., Muga, I., Roggendorf, S., & van der Zee, K. (2019). The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method. Computational Methods in Applied Mathematics, 19(3), 503-522. https://doi.org/10.1515/cmam-2018-0198

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H10(Ω), the Banach Sobolev space W1,q0(Ω), 1 less than ∞ , is more general allowing more irregular solutions. In this paper we present a... Read More about The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method.

Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods (2018)
Journal Article
Antonietti, P., Houston, P., & Pennesi, G. (2018). Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods. Journal of Scientific Computing, 77(3), 1339-1370. https://doi.org/10.1007/s10915-018-0802-y

In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tet... Read More about Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods.

Adaptive refinement for hp-version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem (2018)
Journal Article
Congreve, S., Houston, P., & Perugia, I. (2019). Adaptive refinement for hp-version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem. Advances in Computational Mathematics, 45(1), 361-393. https://doi.org/10.1007/s10444-018-9621-9

In this article we develop an hp-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis enrichment (p-refin... Read More about Adaptive refinement for hp-version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem.

Automatic symbolic computation for discontinuous Galerkin finite element methods (2018)
Journal Article
Houston, P., & Sime, N. (2018). Automatic symbolic computation for discontinuous Galerkin finite element methods. SIAM Journal on Scientific Computing, 40(3), Article C327-C357. https://doi.org/10.1137/17M1129751

The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power... Read More about Automatic symbolic computation for discontinuous Galerkin finite element methods.

Output feedback control of flow separation over an aerofoil using plasma actuators (2018)
Journal Article
Broglia, R., Choi, K., Houston, P., Pasquale, L., & Zanchetta, P. (2018). Output feedback control of flow separation over an aerofoil using plasma actuators. International Journal of Numerical Analysis and Modeling, 15(6),

We address the problem of controlling the unsteady flow separation over an aerofoil, using plasma actuators. Despite the complexity of the dynamics of interest, we show how the problem of controlling flow separation can be formulated as a simple set... Read More about Output feedback control of flow separation over an aerofoil using plasma actuators.

An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems (2018)
Journal Article
Houston, P., & Wihler, T. P. (in press). An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems. Mathematics of Computation, https://doi.org/10.1090/mcom/3308

In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-versio... Read More about An hp-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems.

hp-Adaptive discontinuous Galerkin methods for neutron transport criticality problems (2017)
Journal Article
Hall, E., Houston, P., & Murphy, S. (in press). hp-Adaptive discontinuous Galerkin methods for neutron transport criticality problems. SIAM Journal on Scientific Computing, 39(5), Article B916-B942

In this article we consider the application of high-order/hp-version adaptive discontinuous Galerkin finite element methods (DGFEMs) for the discretization of the keff-eigenvalue problem associated with the neutron transport equation. To this end, we... Read More about hp-Adaptive discontinuous Galerkin methods for neutron transport criticality problems.

An adaptive variable order quadrature strategy (2017)
Journal Article
Houston, P., & Wihler, T. P. (in press). An adaptive variable order quadrature strategy. Lecture Notes in Computational Science and Engineering, 119, https://doi.org/10.1007/978-3-319-65870-4_38

In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this way we aim t... Read More about An adaptive variable order quadrature strategy.

Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method (2017)
Journal Article
Houston, P., & Sime, N. (2017). Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method. Journal of Physics D: Applied Physics, 50(29), Article 295202. https://doi.org/10.1088/1361-6463/aa77dc

In this article we develop a fully self consistent mathematical model describing the formation of a hydrogen plasma in a microwave power assisted chemical vapour deposition (MPA-CVD) reactor employed for the manufacture of synthetic diamond. The unde... Read More about Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method.

Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes (2017)
Journal Article
Antonietti, P. F., Houston, P., Hu, X., Sarti, M., & Verani, M. (in press). Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes. Numerische Mathematik, 54(4), https://doi.org/10.1007/s10092-017-0223-6

In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of s... Read More about Multigrid algorithms for hp-version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes.

Adjoint error estimation and adaptivity for hyperbolic problems (2017)
Book Chapter
Houston, P. (2017). Adjoint error estimation and adaptivity for hyperbolic problems. In R. Abgrall, & C. Shu (Eds.), Handbook of Numerical Methods for Hyperbolic Problems. Applied and Modern Issues. Elsevier / North Holland

In this article we present an overview of a posteriori error estimation and adaptive mesh design for hyperbolic/nearly-hyperbolic problems. In particular, we discuss the question of error estimation for general target functionals of the solution; typ... Read More about Adjoint error estimation and adaptivity for hyperbolic problems.

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains (2016)
Book Chapter
Antonietti, P. F., Cangiani, A., Collis, J., Dong, Z., Georgoulis, E. H., Giani, S., & Houston, P. (2016). Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains. In G. R. Barrenechea, F. Brezzi, A. Cangiani, & E. H. Georgoulis (Eds.), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (281-310). Cham: Springer Publishing Company. https://doi.org/10.1007/978-3-319-41640-3_9

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of st... Read More about Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains.

Adaptive discontinuous Galerkin methods on polytopic meshes (2016)
Conference Proceeding
Collis, J., & Houston, P. (2016). Adaptive discontinuous Galerkin methods on polytopic meshes.

In this article we consider the application of discontinuous Galerkin finite element methods, defined on agglomerated meshes consisting of general polytopic elements, to the numerical approximation of partial differential equation problems posed on c... Read More about Adaptive discontinuous Galerkin methods on polytopic meshes.

hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes (2016)
Journal Article
Cangiani, A., Dong, Z., Georgoulis, E. H., & Houston, P. (2016). hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), 699-725. https://doi.org/10.1051/m2an/2015059

We consider the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopi... Read More about hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes.

Adaptive energy minimisation for hp-finite element methods (2016)
Journal Article
Houston, P., & Wihler, T. P. (2016). Adaptive energy minimisation for hp-finite element methods. Computers and Mathematics with Applications, 71(4), https://doi.org/10.1016/j.camwa.2016.01.002

This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adapti... Read More about Adaptive energy minimisation for hp-finite element methods.

Flows of granular material in two-dimensional channels (2015)
Journal Article
Bain, O., Billingham, J., Houston, P., & Lowndes, I. (2015). Flows of granular material in two-dimensional channels. Journal of Engineering Mathematics, https://doi.org/10.1007/s10665-015-9810-1

Secondary cone-type crushing machines are an important part of the aggregate production process. These devices process roughly crushed material into aggregate of greater consistency and homogeneity. We apply a continuum model for granular materials (... Read More about Flows of granular material in two-dimensional channels.

A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods (2015)
Journal Article
Antonietti, P. F., Houston, P., & Smears, I. A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods. Manuscript submitted for publication

In this article, we consider the derivation of hp-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial dif... Read More about A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods.

Goal-oriented a posteriori error estimation for the travel time functional in porous media flows (2015)
Journal Article
Cliffe, A., Collis, J., & Houston, P. (2015). Goal-oriented a posteriori error estimation for the travel time functional in porous media flows. SIAM Journal on Scientific Computing, 37(2), Article B127-B152. https://doi.org/10.1137/140960499

In this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows. The key application of this work is in the safety assessment of ra... Read More about Goal-oriented a posteriori error estimation for the travel time functional in porous media flows.

Goal-oriented adaptive composite discontinuous Galerkin methods for incompressible flows (2014)
Journal Article
Giani, S., & Houston, P. (2014). Goal-oriented adaptive composite discontinuous Galerkin methods for incompressible flows. Journal of Computational and Applied Mathematics, 270, https://doi.org/10.1016/j.cam.2014.03.007

In this article we consider the application of goal-oriented mesh adaptation to problems posed on complicated domains which may contain a huge number of local geometrical features, or micro-structures. Here, we exploit the composite variant of the di... Read More about Goal-oriented adaptive composite discontinuous Galerkin methods for incompressible flows.

hp-adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains (2014)
Journal Article
Giani, S., & Houston, P. (2014). hp-adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains. Numerical Methods for Partial Differential Equations, 30(4), https://doi.org/10.1002/num.21872

In this paper we develop the a posteriori error estimation of hp-version discontinuous Galerkin composite finite element methods for the discretization of second order elliptic partial differential equations. This class of methods allows for the appr... Read More about hp-adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains.

Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains (2014)
Journal Article
Antonietti, P. F., Giani, S., & Houston, P. (2014). Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains. Journal of Scientific Computing, 60(1), https://doi.org/10.1007/s10915-013-9792-y

In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized... Read More about Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains.

Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows (2014)
Journal Article
Giani, S., & Houston, P. (2014). Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows. Numerical Mathematics, 7(2), https://doi.org/10.1017/S100489790000091X

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, w... Read More about Domain decomposition preconditioners for discontinuous Galerkin discretizations of compressible fluid flows.

hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes (2014)
Journal Article
Cangiani, A., Georgoulis, E. H., & Houston, P. (2014). hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences, 24(10), 2009-2041. https://doi.org/10.1142/S0218202514500146

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analy... Read More about hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes.

Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows (2014)
Journal Article
Congreve, S., & Houston, P. (2014). Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows. International Journal of Numerical Analysis and Modeling, 11(3),

In this article we consider the a priori and a posteriori error analysis of two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a strongly monotone quasi-Newtonian fluid flow problem. The basis of the two-g... Read More about Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows.

hp-Adaptive discontinuous Galerkin methods for bifurcation phenomena in open flows (2013)
Journal Article
Cliffe, A., Hall, E., & Houston, P. (2014). hp-Adaptive discontinuous Galerkin methods for bifurcation phenomena in open flows. Computers and Mathematics with Applications, 67(4), https://doi.org/10.1016/j.camwa.2013.09.024

In this article we consider the a posteriori error estimation and adaptive mesh refinement of hp-version discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations... Read More about hp-Adaptive discontinuous Galerkin methods for bifurcation phenomena in open flows.

hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains (2013)
Journal Article
Antonietti, P. F., Giani, S., & Houston, P. (2013). hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM Journal on Scientific Computing, 35(3), Article A1417-A1439. https://doi.org/10.1137/120877246

In this paper we introduce the hp-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on c... Read More about hp-Version composite discontinuous Galerkin methods for elliptic problems on complicated domains.

Is a persistent global bias necessary for the establishment of planar cell polarity? (2013)
Journal Article
Fischer, S., Houston, P., Monk, N. A., & Owen, M. R. (2013). Is a persistent global bias necessary for the establishment of planar cell polarity?. PLoS ONE, 8(4), 1-12. https://doi.org/10.1371/journal.pone.0060064

Planar cell polarity (PCP) — the coordinated polarisation of a whole field of cells within the plane of a tissue — relies on the interaction of three modules: a global module that couples individual cellular polarity to the tissue axis, a local modul... Read More about Is a persistent global bias necessary for the establishment of planar cell polarity?.

hp-adaptive two-grid discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows (2013)
Conference Proceeding
Congreve, S., Houston, P., & Wihler, T. P. (2013). hp-adaptive two-grid discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows.

We develop the a posteriori error analysis, with respect to a mesh-dependent energy norm, of two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian flows. The performance of the proposed estimators within an hp-adaptive... Read More about hp-adaptive two-grid discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows.

Application of hp-adaptive discontinuous Galerkin methods to bifurcation phenomena in pipe flows (2013)
Conference Proceeding
Cliffe, A., Hall, E., & Houston, P. (2013). Application of hp-adaptive discontinuous Galerkin methods to bifurcation phenomena in pipe flows.

In this article we consider the a posteriori error estimation and adaptive mesh refinement of hp-version discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations... Read More about Application of hp-adaptive discontinuous Galerkin methods to bifurcation phenomena in pipe flows.

Two-grid hp-DGFEM for second order quasilinear elliptic PDEs based on an incomplete Newton iteration (2013)
Conference Proceeding
Congreve, S., & Houston, P. (2013). Two-grid hp-DGFEM for second order quasilinear elliptic PDEs based on an incomplete Newton iteration.

In this paper we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem based on the application of a single step of a no... Read More about Two-grid hp-DGFEM for second order quasilinear elliptic PDEs based on an incomplete Newton iteration.

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows (2012)
Journal Article
Congreve, S., Houston, P., Süli, E., & Wihler, T. P. Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows. Manuscript submitted for publication

In this article we develop both the a priori and a posteriori error analysis of hp– version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ R^d, d = 2,... Read More about Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows.

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry (2011)
Journal Article
Cliffe, A., Hall, E., Houston, P., Phipps, E., & Salinger, A. (2012). Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry. Journal of Scientific Computing, 52(1), 153-179. https://doi.org/10.1007/s10915-011-9545-8

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particula... Read More about Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry.

Two-grid hp-version DGFEMs for strongly monotone second-order quasilinear elliptic PDEs (2011)
Journal Article
Congreve, S., Houston, P., & Wihler, T. P. (2011). Two-grid hp-version DGFEMs for strongly monotone second-order quasilinear elliptic PDEs. PAMM, 11(1), https://doi.org/10.1002/pamm.201110002

In this article we develop the a priori error analysis of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of strongly monotone second-order quasilinear partial differential equations. In thi... Read More about Two-grid hp-version DGFEMs for strongly monotone second-order quasilinear elliptic PDEs.

Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows (2011)
Journal Article
Giani, S., & Houston, P. Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows. Manuscript submitted for publication

In this article we consider the construction of general isotropic and anisotropic adaptive mesh refinement strategies, as well as hp-mesh refinement techniques, for the numerical approximation of the compressible Euler and Navier-Stokes equations. To... Read More about Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows.

Discontinuous Galerkin methods for problems with Dirac delta source (2011)
Journal Article
Houston, P., & Wihler, T. P. Discontinuous Galerkin methods for problems with Dirac delta source. Manuscript submitted for publication

In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-un... Read More about Discontinuous Galerkin methods for problems with Dirac delta source.

Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem (2010)
Journal Article
Cliffe, A., Hall, E., Houston, P., Phipps, E. T., & Salinger, A. G. (2010). Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem. Communications in Computational Physics, 8(4), 845-865. https://doi.org/10.4208/cicp.290709.120210a

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied. In particular, we study in detail the numerical approximation of the Bratu problem, base... Read More about Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem.

Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry (2010)
Journal Article
Cliffe, A., Hall, E., Houston, P., Phipps, E. T., & Salinger, A. G. Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry. Manuscript submitted for publication

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particula... Read More about Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry.

A new method for conditioning stochastic groundwater flow models in fractured media (2010)
Journal Article
Milne, A., Cliffe, A., Holton, D., Houston, P., Jackson, C. P., & Joyce, S. A new method for conditioning stochastic groundwater flow models in fractured media. Manuscript submitted for publication

Many geological formations consist of crystalline rocks that have very low matrix permeability but allow flow through an interconnected network of fractures. Understanding the flow of groundwater through such rocks is important in considering disposa... Read More about A new method for conditioning stochastic groundwater flow models in fractured media.

High-order hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows (2010)
Book Chapter
Giani, S., & Houston, P. (2010). High-order hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows. In N. Kroll, H. Bieler, H. Deconinck, V. Couallier, H. van der Ven, & K. Sorensen (Eds.), ADIGMA - a European initiative on the development of adaptive higher-order variational methods for aerospace applications. Springer. https://doi.org/10.1007/978-3-642-03707-8_28

This article is concerned with the construction of general isotropic and anisotropic adaptive strategies, as well as hp-mesh refinement techniques, in combination with dual-weighted-residual a posteriori error indicators for the discontinuous Galerki... Read More about High-order hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows.

Second-order elliptic PDE with discontinuous boundary data (2009)
Journal Article
Houston, P., & Wihler, T. P. Second-order elliptic PDE with discontinuous boundary data. Manuscript submitted for publication

We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point... Read More about Second-order elliptic PDE with discontinuous boundary data.

A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods (2009)
Journal Article
Antonietti, P. F., & Houston, P. A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. Manuscript submitted for publication

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In... Read More about A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods.

An hr-adaptive discontinuous Galerkin method for advection-diffusion problems (2009)
Journal Article
Antonietti, P. F., & Houston, P. An hr-adaptive discontinuous Galerkin method for advection-diffusion problems. Manuscript submitted for publication

We propose an adaptive mesh refinement strategy based on exploiting a combination of a pre-processing mesh re-distribution algorithm employing a harmonic mapping technique, and standard (isotropic) mesh subdivision for discontinuous Galerkin approxim... Read More about An hr-adaptive discontinuous Galerkin method for advection-diffusion problems.

Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions (2009)
Journal Article
Zhu, L., Giani, S., Houston, P., & Schoetzau, D. Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions. Manuscript submitted for publication

We develop the energy norm a-posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a... Read More about Energy norm a-posteriori error estimation for hp-adaptive discontinuous Galerkin methods for elliptic problems in three dimensions.

Modelling and analysis of planar cell polarity (2008)
Journal Article
Schamberg, S., Houston, P., Monk, N. A., & Owen, M. R. Modelling and analysis of planar cell polarity. Manuscript submitted for publication

Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles and feathers; physiologically, it can organise ciliary beating. Here we present two approaches to modelling this phenomenon. The aim i... Read More about Modelling and analysis of planar cell polarity.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity
Journal Article
Georgoulis, E. H., Hall, E., & Houston, P. Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity. Manuscript submitted for publication

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined co... Read More about Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity.

Error estimation and adaptive mesh refinement for aerodynamic flows
Book Chapter
Hartmann, R., & Houston, P. Error estimation and adaptive mesh refinement for aerodynamic flows. In H. Deconinck (Ed.), Proceedings of the 36THCFD/Adigma course on HP-adaptive and HP-multigrid methods. Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium: von Karman Institute for Fluid Dynamics

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compress... Read More about Error estimation and adaptive mesh refinement for aerodynamic flows.

Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows
Journal Article
Cliffe, A., Hall, E., & Houston, P. Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. Manuscript submitted for publication

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Parti... Read More about Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows.

An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems
Journal Article
Georgoulis, E. H., Houston, P., & Virtanen, J. An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems. Manuscript submitted for publication

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approxi... Read More about An A Posteriori Error Indicator for Discontinuous Galerkin Approximations of Fourth Order Elliptic Problems.

A Mixed Discontinuous Galerkin Method for Incompressible Magnetohydrodynamics
Journal Article
Houston, P., Schoetzau, D., & Wei, X. A Mixed Discontinuous Galerkin Method for Incompressible Magnetohydrodynamics. Manuscript submitted for publication

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous P^3_{k}-P_{k-1} elements... Read More about A Mixed Discontinuous Galerkin Method for Incompressible Magnetohydrodynamics.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes
Journal Article
Georgoulis, E. H., Hall, E., & Houston, P. (2006). Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational me... Read More about Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis
Journal Article
Georgoulis, E. H., Hall, E., & Houston, P. (2006). Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite... Read More about Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis.

An Optimal Order Interior Penalty Discontinuous Galerkin Discretization of the Compressible Navier-Stokes Equations
Journal Article
Hartmann, R., & Houston, P. An Optimal Order Interior Penalty Discontinuous Galerkin Discretization of the Compressible Navier-Stokes Equations. Manuscript submitted for publication

In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construct... Read More about An Optimal Order Interior Penalty Discontinuous Galerkin Discretization of the Compressible Navier-Stokes Equations.

Discontinuous Galerkin Methods for the Biharmonic Problem
Journal Article
Georgoulis, E. H., & Houston, P. Discontinuous Galerkin Methods for the Biharmonic Problem. Manuscript submitted for publication

This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems involving the biharmonic operator. The first part extends the unified approach of Arnold, Brezzi, Cockbur... Read More about Discontinuous Galerkin Methods for the Biharmonic Problem.

A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems
Journal Article
Houston, P., Suli, E., & Wihler, T. P. (2006). A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error ar... Read More about A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems.

Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs
Journal Article
Congreve, S., Houston, P., & Wihler, T. P. Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs. Manuscript submitted for publication

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting... Read More about Two-grid hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic PDEs.

A Pre-processing Moving Mesh Method for Discontinuous Galerkin Approximations of Advection-Diffusion-Reaction Problems
Journal Article
Antonietti, P. F., & Houston, P. A Pre-processing Moving Mesh Method for Discontinuous Galerkin Approximations of Advection-Diffusion-Reaction Problems. Manuscript submitted for publication

We propose a pre-processing mesh re-distribution algorithm based upon harmonic maps employed in conjunction with discontinuous Galerkin approximations of advection-diffusion-reaction problems. Extensive two-dimensional numerical experiments with dif... Read More about A Pre-processing Moving Mesh Method for Discontinuous Galerkin Approximations of Advection-Diffusion-Reaction Problems.

A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations
Journal Article
discretizations of H(curl)-elliptic partial differential equations

We develop the a posteriori error estimation of interior penalty discontinuous Galerkin discretizations for H(curl)-elliptic problems that arise in eddy current models. Computable upper and lower bounds on the error measured in terms of a natural (me... Read More about A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations.

Discontinuous Galerkin Computation of the Maxwell Eigenvalues on Simplicial Meshes
Journal Article
Buffa, A., Houston, P., & Perugia, I. (2005). Discontinuous Galerkin Computation of the Maxwell Eigenvalues on Simplicial Meshes

This paper is concerned with the discontinuous Galerkin approximation of the Maxwell eigenproblem. After reviewing the theory developed in [5], we present a set of numerical experiments which both validate the theory, and provide further insight reg... Read More about Discontinuous Galerkin Computation of the Maxwell Eigenvalues on Simplicial Meshes.

Enhancing SPH using moving least-squares and radial basis functions
Journal Article
Brownlee, R., Houston, P., Levesley, J., & Rosswog, S. (2005). Enhancing SPH using moving least-squares and radial basis functions

In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using... Read More about Enhancing SPH using moving least-squares and radial basis functions.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation
Journal Article
Hartmann, R., & Houston, P. (2005). Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation

In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier--Stokes equations. In particular... Read More about Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation.

Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation
Journal Article
Navier-Stokes Equations I: Method Formulation

In this article we consider the development of discontinuous Galerkin finite element methods for the numerical approximation of the compressible Navier-Stokes equations. For the discretization of the leading order terms, we propose employing the gene... Read More about Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations I: Method Formulation.