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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.