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PAUL HOUSTON

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PAUL HOUSTON

Professor of Computational and Applied Maths


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

Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation (2020)
Journal Article
Houston, P., Roggendorf, S., & G. van der Zee, K. (2020). Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation. Computers and Mathematics with Applications, 80(5), 851-873. https://doi.org/10.1016/j.camwa.2020.03.025

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can... Read More about Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation.

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. doi: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.