arXiv subjects

Paul Houston

Publications and source records attributed to Paul Houston.

At least 19 recordsLinked to original sources

arXiv2024

Gradient Flow Finite Element Discretisations with Energy-Based $hp$-Adaptivity for the Gross-Pitaevskii Equation with Angular Momentum

This article deals with the stationary Gross-Pitaevskii non-linear eigenvalue problem in the presence of a rotating magnetic field that is used to model macroscopic quantum effects such as Bose-Einstein condensates (BECs). In this regime, the ground-state wave-function can exhibit an a priori unknown number of quantum vortices at unknown locations, which necessitates the exploitation of adaptive numerical strategies. To this end, we consider the conforming finite element method in combination with a discrete Sobolev gradient descent, which is guided by the energy-topology of the problem, to address the nonlinearity. In addition, a key novelty of this work is an $hp$-adaptive strategy that is solely based on energy decay rather than a posteriori error estimators for the refinement process. Numerical results demonstrate that the $hp$-adaptive strategy is highly efficient in terms of accuracy to compute the ground-state wave function and energy for several test problems where we observe exponential convergence.

math.NA↗

arXiv2024

Can We Learn the Energy of Sublimation of Ice from Water Clusters?

This short paper reports a study of the electronic dissociation energies, De, of water clusters from direct ab initio (mostly CCSD(T)) calculations and the q-AQUA and MB-pol potentials. These clusters range in size from 6-25 monomers. These are all in very good agreement with each other, as shown in a recent Perspective by Herman and Xantheas. To the best of our knowledge, we present for the first time results for the De per monomer. To our surprise this quantity appears to be converging to a value close to 12 kcal/mol. An estimate of 1.5 - 2 kcal/mol for the {\Delta}ZPE for these clusters puts the value of D0 at 10 to 10.5 kcal/mol. This value is remarkably (and probably fortuitously) close to the reported sublimation enthalpy of 10.2 kcal/mol at 10 K. However, given that these De energies correspond to dissociation of the cluster to N isolated monomers the interpretation of ``vaporization" of these ``solid" clusters is qualitatively reasonable.

physics.chem-ph↗

arXiv2024

Exponential Convergence of $hp$-ILGFEM for semilinear elliptic boundary value problems with monomial reaction

We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\Omega\subset\mathbb{R}^2$ with a finite number of straight edges. In particular, we analyze the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\Omega$, with geometric corner refinement, with polynomial degrees increasing in sync with the geometric mesh refinement towards the corners of $\Omega$. For a sequence of discrete solutions generated by the ILG solver, with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution, we prove exponential convergence in $\mathrm{H}^1(\Omega)$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.

math.NA↗

arXiv2023

Analysis of Iterative Methods for the Linear Boltzmann Transport Equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the discretisation parameters. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport.

math.NA↗

arXiv2023

Quadrature-Free Polytopic Discontinuous Galerkin Methods for Transport Problems

In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two- and three-dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.

math.NA↗

arXiv2023

A divergence free $C^0$-RIPG stream function formulation of the incompressible Stokes system with variable viscosity

Pointwise divergence free velocity field approximations of the Stokes system are gaining popularity due to their necessity in precise modelling of physical flow phenomena. Several methods have been designed to satisfy this requirement; however, these typically come at a greater cost when compared with standard conforming methods, for example, because of the complex implementation and development of specialized finite element bases. Motivated by the desire to mitigate these issues for 2D simulations, we present a $C^0$-interior penalty Galerkin (IPG) discretization of the Stokes system in the stream function formulation. In order to preserve a spatially varying viscosity this approach does not yield the standard and well known biharmonic problem. We further employ the so-called robust interior penalty Galerkin (RIPG) method; stability and convergence analysis of the proposed scheme is undertaken. The former, which involves deriving a bound on the interior penalty parameter is particularly useful to address the $\mathcal{O}(h^{-4})$ growth in the condition number of the discretized operator. Numerical experiments confirming the optimal convergence of the proposed method are undertaken. Comparisons with thermally driven buoyancy mantle convection model benchmarks are presented.

math.NA↗

arXiv2023

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an $hp$-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and $hp$-version a priori error analysis of the proposed method, by deriving suitable $hp$-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

math.NA↗

arXiv2021

Two-grid $hp$-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

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 polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid $hp$-version discontinuous Galerkin finite element method from 10.1007/s10915-012-9644-1 for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an $hp$-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed $hp$-adaptive two-grid method.

math.NA↗

arXiv2021

Linearisation of the Travel Time Functional in Porous Media Flows

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 post-closure safety assessment of deep geological storage facilities for radioactive waste where leaked, non-sorbing, solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented $a$ $posteriori$ error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its G\^ateaux derivative. A mixed finite element method is implemented to approximate Darcy's equations and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, in the UK.

math.NA↗

arXiv2021

Breaking the Coupled Cluster Barrier for Machine Learned Potentials of Large Molecules: The Case of 15-atom Acetylacetone

Machine-learned potential energy surfaces (PESs) for molecules with more than 10 atoms are typically forced to use lower-level electronic structure methods such as density functional theory and second-order Moller-Plesset perturbation theory (MP2). While these are efficient and realistic, they fall short of the accuracy of the ``gold standard'' coupled-cluster method, especially with respect to reaction and isomerization barriers. We report a major step forward in applying a $\Delta$-machine learning method to the challenging case of acetylacetone, whose MP2 barrier height for H-atom transfer is low by roughly 1.5 kcal/mol relative to the benchmark CCSD(T) barrier of 3.2 kcal/mol. From a database of 2151 local CCSD(T) energies, and training with as few as 430 energies, we obtain a new PES with a barrier of 3.49 kcal/mol in agreement with the LCCSD(T) one of 3.54 kcal/mol and close to the benchmark value. Tunneling splittings due to H-atom transfer are calculated using this new PES, providing improved estimates over previous ones obtained using an MP2-based PES.

physics.chem-ph↗

arXiv2020

$\Delta$-Machine Learning for Potential Energy Surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) Level of Theory

``$\Delta$-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients to close to a coupled cluster (CC) level of accuracy. Here we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation $V_{LL{\rightarrow}CC}=V_{LL}+\Delta{V_{CC-LL}}$, and demonstrated for \ce{CH4}, \ce{H3O+}, and $trans$ and $cis$-$N$-methyl acetamide (NMA), \ce{CH3CONHCH3}. For these molecules, the LL PES, $V_{LL}$, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients, and $\Delta{V_{CC-LL}}$ is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For \ce{CH4} these are new calculations adopting an aug-cc-pVDZ basis, for \ce{H3O+} previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA training is done with 4696 CCSD(T) energies.

physics.chem-ph↗

arXiv2019

Gibbs Phenomena for $L^q$-Best Approximation in Finite Element Spaces -- Some Examples

Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as $q$ tends to $1$. The aim here is to show the potential of $L^1$ as a solution space in connection with suitably designed meshes.

math.NA↗

arXiv2019

Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation

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 lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < $\infty$. We then apply a non-standard, non-linear PetrovGalerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.

math.NA↗

arXiv2018

Automatic symbolic computation for discontinuous Galerkin finite element methods

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 series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Fr\'echet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems.

math.NA↗

arXiv2017

Adaptive Refinement for $hp$-Version Trefftz Discontinuous Galerkin Methods for the Homogeneous Helmholtz Problem

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-refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples.

math.NA↗

arXiv2016

An $hp$-Adaptive Newton-Discontinuous-Galerkin Finite Element Approach for Semilinear Elliptic Boundary Value Problems

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 $hp$-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust $hp$-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

math.NA↗

arXiv2015

An Adaptive Variable Order Quadrature Strategy

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 to account for local smoothness properties of the function to be integrated as effectively as possible, and thereby achieve highly accurate results in a very efficient manner. Indeed, this idea originates from so-called hp-version finite element methods which are known to deliver high-order convergence rates, even for nonsmooth functions.

math.NA↗

arXiv2015

Adaptive Energy Minimisation for $hp$-Finite Element Methods

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 adaptive manner. Specifically, we outline a new approach in the context of $hp$-adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second-order boundary value problems. Numerical experiments are presented which highlight the practical performance of this new $hp$-refinement technique for both one- and two-dimensional problems.

math.NA↗