Related papers: Extending FEniCS to Work in Higher Dimensions Usin…
Friedrichs' systems (FS) are symmetric positive linear systems of first-order partial differential equations (PDEs), which provide a unified framework for describing various elliptic, parabolic and hyperbolic semi-linear PDEs such as the…
In this paper, we consider the Poisson equation on a "long" domain which is the Cartesian product of a one-dimensional long interval with a (d-1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will…
In this article, we present an Unfitted Space-Time Finite Element method for the scalar transport equation posed on moving domains. We consider the case of the domain boundary being transported by the same velocity field as the scalar…
We propose a high order unfitted finite element method for solving timeharmonic Maxwell interface problems. The unfitted finite element method is based on a mixed formulation in the discontinuous Galerkin framework on a Cartesian mesh with…
We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general…
Finite element method is one of powerful numerical methods to solve PDE. Usually, if a finite element solution to a Poisson equation based on a triangulation of the underlying domain is not accurate enough, one will discard the solution and…
We propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so-called TFETI (Total Finite Element Tearing and Interconnecting) domain decomposition method. We consider an…
The novel idea of weak Galerkin (WG) finite element methods is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees.…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
We study a numerical reconstruction strategy for the potential in the fractional Calder\'on problem from a single partial exterior measurement. The forward model is the fractional Schr\"odinger equation in a bounded domain, with prescribed…
Fourier solvers have become efficient tools to establish structure-property relations in heterogeneous materials. Introduced as an alternative to the Finite Element (FE) method, they are based on fixed-point solutions of the…
High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the…
In this paper, we develop a framework for solving inverse deformation problems using the FEniCS Project finite element software. We validate our approach with experimental imaging data acquired from a soft silicone beam under gravity. In…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new…
This paper is concerned with the development of weak Galerkin (WG) finite element method for optimal control problems governed by second order elliptic partial differential equations (PDEs). It is advantageous to use discontinuous finite…
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different…