arXiv subjects

Matthew W. Scroggs

Publications and source records attributed to Matthew W. Scroggs.

8 recordsLinked to original sources

arXiv2026

Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices

The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since on these architectures performance at low degree is limited (severely) by the available memory bandwidth and only a very small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power of modern architectures. However, whilst stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We address this with a new algorithm that can be used in computing degree-of-freedom maps for an arbitrary Ciarlet-type finite element using only the element's definition and properties of the reference cell, and without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm allows for new elements to be implemented with ease and has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.

math.NA↗

arXiv2025

DefElement: an encyclopedia of finite element definitions

DefElement is an online encyclopedia of finite element definitions that was created and is maintained by the authors of this paper. DefElement aims to make information about elements defined in the literature easily available in a standard format. There are a number of open-source finite element libraries available, and it can be difficult to check that an implementation of an element in a library matches the element's definition in the literature or implementation in another library, especially when many libraries include variants of elements whose basis functions do not match exactly. In this paper, we carefully derive conditions under which elements can be considered equivalent and describe an algorithm that uses these conditions to verify that two implementations of a finite element are indeed variants of the same element. The results of scheduled runs of our implementation of this verification algorithm are included in the information available on the DefElement website.

math.NA↗

arXiv2023

Coupling finite and boundary element methods to solve the Poisson--Boltzmann equation for electrostatics in molecular solvation

The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to the accurate representation of the molecular surface and partial charges, and exact enforcement of the boundary conditions at infinity. However, the boundary element method is limited to linear equations and piecewise constant variations of the material properties. In this work, we present a scheme that couples finite and boundary elements for the Poisson--Boltzmann equation, where the finite element method is applied in a confined {\it solute} region, and the boundary element method in the external {\it solvent} region. As a proof-of-concept exercise, we use the simplest methods available: Johnson--Nédélec coupling with mass matrix and diagonal preconditioning, implemented using the Bempp-cl and FEniCSx libraries via their Python interfaces. We showcase our implementation by computing the polar component of the solvation free energy of a set of molecules using a constant and a Gaussian-varying permittivity. We validate our implementation against the finite difference code APBS (to 0.5\%), and show scaling from protein G B1 (955 atoms) up to immunoglobulin G (20\,148 atoms). For small problems, the coupled method was efficient, outperforming a purely boundary integral approach. For Gaussian-varying permittivities, which are beyond the applicability of boundary elements alone, we were able to run medium to large sized problems on a single workstation. Development of better preconditioning techniques and the use of distributed memory parallelism for larger systems remains an area for future work. We hope this work will serve as inspiration for future developments for molecular electrostatics with implicit solvent models.

physics.comp-ph↗

arXiv2022

Boundary element methods for Helmholtz problems with weakly imposed boundary conditions

We consider boundary element methods where the Calderón projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet conditions on the Helmholtz equation, and extend the analysis of the Laplace problem from \emph{Boundary element methods with weakly imposed boundary conditions} to this case. The theory is illustrated by a series of numerical examples.

math.NA↗

arXiv2022

Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes

We develop a method for generating degree-of-freedom maps for arbitrary order finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and in the case of hexahedral cells not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including Lagrange, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in an open-source finite element library.

math.NA↗

arXiv2020

Weak imposition of Signorini boundary conditions on the boundary element method

We derive and analyse a boundary element formulation for boundary conditions involving inequalities. In particular, we focus on Signorini contact conditions. The Calderón projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. We present a complete numerical a priori error analysis and present some numerical examples to illustrate the theory.

math.NA↗

arXiv2018

Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

We consider the simulation of electromagnetic scattering by single and multiple isotropic homogeneous dielectric particles using boundary integral equations. Galerkin discretizations of the classical Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) boundary integral equation formulation provide accurate solutions for complex particle geometries, but are well-known to lead to ill-conditioned linear systems. In this paper we carry out an experimental investigation into the performance of Calderón preconditioning techniques for single and multiple absorbing obstacles, which involve a squaring of the PMCHWT operator to produce a well-conditioned second-kind formulation. For single-particle scattering configurations we find that Calderón preconditioning is actually often outperformed by simple "mass-matrix" preconditioning, i.e. working with the strong form of the discretized PMCHWT operator. In the case of scattering by multiple particles we find that a significant saving in computational cost can be obtained by performing block-diagonal Calderón preconditioning in which only the self-interaction blocks are preconditioned. Using the boundary element software library Bempp (www.bempp.com) the numerical performance of the different methods is compared for a range of wavenumbers, particle geometries and complex refractive indices relevant to the scattering of light by atmospheric ice crystals.

math.NA↗

arXiv2018

Boundary element methods with weakly imposed boundary conditions

We consider boundary element methods where the Calderón projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet, mixed Dirichlet--Neumann, and Robin conditions. A salient feature of the Robin condition is that the conditioning of the system is robust also for stiff boundary conditions. The theory is illustrated by a series of numerical examples.

math.NA↗