Boundary elements with mesh refinements for the wave equation

Computer Methods in Applied Mechanics and Engineering

Stocek

2019

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This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space-time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for 2dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space-time adaptive, as well as uniform and graded discretizations.

Section: Time-independent Problemssupporting

confidence: 81%

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

Computer Methods in Applied Mechanics and Engineering

Stocek

2019

Self Cite

“…Together with the a priori estimates for the time domain boundary element methods on screens [23,24], our results imply convergence rates for the p-version Galerkin approximations which are twice those observed for the quasi-uniform h-method in [22].…”

supporting

confidence: 57%

“…From [22], the convergence rate in energy norm of the uniform h-method on the screen is 0.5 as h tends to 0. A cross section at y = 0 of the solution for the right hand side f 1 (t, (x, y, z) T ) = sin 5 (t)x 2 is shown in Figure 3, for a uniform triangulation of Γ with 1250 triangles at times t = 1.0 and 1.4.…”

Section: Wave Equation Outside a Screenmentioning

confidence: 98%

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hp-version time domain boundary elements for the wave equation on quasi-uniform meshes

Computer Methods in Applied Mechanics and Engineering

Özdemir

Stark

et al. 2019

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Solutions to the wave equation in the exterior of a polyhedral domain or a screen in R 3 exhibit singular behavior from the edges and corners. We present quasi-optimal hp-explicit estimates for the approximation of the Dirichlet and Neumann traces of these solutions for uniform time steps and (globally) quasi-uniform meshes on the boundary. The results are applied to an hp-version of the time domain boundary element method. Numerical examples confirm the theoretical results for the Dirichlet problem both for screens and polyhedral domains.This article initiates the study of high-order boundary elements in the time domain. For elliptic problems, p-and hp-versions of the finite element method give rise to fast approximations of both smooth solutions and geometric singularities. These methods converge to the solution by increasing the polynomial degree p of the elements, possibly in combination with reducing the mesh size h of the quasi-uniform mesh. They were first investigated in the group of Babuska [3,4,17,18]. See [49] for a comprehensive analysis for 2d problems.The analogous p-and hp-versions of the boundary element method go back to [2,53,54]. More recent optimal convergence results for boundary elements on screens and polyhedral surfaces covering 3d problems have been obtained, for example, in [9,10,11,12,13].Boundary element methods for time dependent problems have recently become of interest [48]. In this article we introduce a space-time hp-version of the time domain boundary element method for the wave equation with non-homogeneous Dirichlet or Neumann boundary conditions.

“…for 1 ≤ n ≤ N t and 1 ≤ j ≤ N s ′ . The resulting discretization of the Poincaré-Steklov operator has been tested in [15,16], and corresponding results are obtained for more natural discretizations with piecewise constant λ h,△t . Piecewise linear and higher order test functions are considered in [19].…”

Section: Numerical Resultsmentioning

confidence: 99%

A time–dependent FEM-BEM coupling method for fluid–structure interaction in 3d

Applied Numerical Mathematics

Özdemir

Stephan

2020

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We consider the well-posedness and a priori error estimates of a 3d FEM-BEM coupling method for fluid-structure interaction in the time domain. For an elastic body immersed in a fluid, the exterior linear wave equation for the fluid is reduced to an integral equation on the boundary involving the Poincaré-Steklov operator. The resulting problem is solved using a Galerkin boundary element method in the time domain, coupled to a finite element method for the Lamé equation inside the elastic body. Based on ideas from the timeindependent coupling formulation, we obtain an a priori error estimate and discuss the implementation of the proposed method. Numerical experiments illustrate the performance of our scheme for model problems.