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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.

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

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.