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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.

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.

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.

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.

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.