Related papers: Concepts for Composing Finite Element Function Spa…

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed…

Finite element methods usually construct basis functions and quadrature rules for multidimensional domains via tensor products of one-dimensional counterparts. While straightforward, this approach results in integration spaces larger than…

Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…

We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…

We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh…

The multimesh finite element method enables the solution of partial differential equations on a computational mesh composed by multiple arbitrarily overlapping meshes. The discretization is based on a continuous--discontinuous function…

In the present work, we investigate the computational efficiency afforded by higher-order finite-element discretization of the saddle-point formulation of orbital-free density functional theory. We first investigate the robustness of viable…

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of…

Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…

We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with…

In this paper a class of higher order finite element methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted finite element approach in which standard Taylor-Hood spaces on an underlying…

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation…

We present an approach to solid-state electronic-structure calculations based on the finite-element method. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed of polynomials, the…

Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…

Code generation based software platforms, such as Firedrake, have become popular tools for developing complicated finite element discretisations of partial differential equations. We extended the code generation infrastructure in Firedrake…

The widespread success of foundation models in natural language processing and computer vision has inspired researchers to extend the concept to scientific machine learning and computational science. However, this position paper argues that…

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general…

We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on evolving surfaces and bulk domains as well…

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…

We present for static density functional theory and time-dependent density functional theory calculations an all-electron method which employs high-order hierarchical finite element bases. Our mesh generation scheme, in which structured…