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In this note we shall introduce a simple, effective numerical method for solving partial differential equations for scalar and vector-valued data defined on surfaces. Even though we shall follow the traditional way to approximate the…
The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method allows a surface to be given implicitly as a zero level of a level set function. A surface equation…
This tutorial teaches parts of the finite element method (FEM), and solves a stochastic partial differential equation (SPDE). The contents herein are considered "known" in the numerics literature, but for statisticians it is very difficult…
We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
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…
This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…
The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
This paper addresses the analysis and numerical assessment of a computational method for solving the Cahn--Hilliard equation defined on a surface. The proposed approach combines the stabilized trace finite element method for spatial…
We analyze a new framework for expressing finite element methods on arbitrarily many intersecting meshes: multimesh finite element methods. The multimesh finite element method, first presented in [40], enables the use of separate meshes to…
A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed…
The finite-element method is a preferred numerical method when electromagnetic fields at high accuracy are to be computed in nano-optics design. Here, we demonstrate a finite-element method using hp-adaptivity on tetrahedral meshes for…
This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for…
Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes…
Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local…
In order to simulate elastic wave propagation in a complex structure with inhomogeneous media, we often need to obtain the propagating eigenmodes of an elastic waveguide. As the waveguide is assumed uniform in one direction, the original…
Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…
A numerical method of solving the problem of acoustic wave radiation in the presence of a rigid scatterer is described. It combines the finite element method and the boundary algebraic equations. In the proposed method, the exterior domain…
To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness…