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A construction of prismatic Hardy space infinite elements to discretize wave equations on unbounded domains $\Omega$ in $H^1_{loc}(\Omega)$, $H_{loc}(curl;\Omega)$ and $H_{loc}(div;\Omega)$ is presented. As our motivation is to solve…
In this paper we construct conforming Virtual Element approximations on domains with curved boundary and/or internal curved interfaces, both in two and three dimensions. Our approach allows to impose both Dirichlet and Neumann…
For the 3-D quadratic quasilinear wave equations in exterior domains with Dirichlet or Neumann boundary conditions, the global existence or the maximal existence time of small data smooth solutions have been established in the past.…
In this paper, we study the exterior problem for the maximal surface equation. We obtain the precise asymptotic behavior of the exterior solution at infinity. And we prove that the exterior Dirichlet problem is uniquely solvable given…
Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on…
This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational…
Here we study the Dirichlet problem for first order linear and quasi-linear hyperbolic PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. By means of a Lyapunov…
The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local…
Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution…
In this paper we study the finite element approximation of systems of second-order nonlinear hyperbolic equations. The proposed numerical method combines a $hp$-version discontinuous Galerkin finite element approximation in the time…
We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet…
We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$…
An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The…
We provide a fully nonlinear port-Hamiltonian formulation for discrete elastodynamical systems as well as a structure-preserving time discretization. The governing equations are obtained in a variational manner and represent index-1…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero…
In \cite{cheung2019optimally}, the authors presented two finite element methods for approximating second order boundary value problems on polytopial meshes with optimal accuracy without having to utilize curvilinear mappings. This was done…
The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a…
A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model…
The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising in self-organised criticality. Based on…
We consider a two-dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where…