Very Weak Space-Time Variational Formulation for the Wave Equation: Analysis and Efficient Numerical Solution

07/26/2021
by   Julian Henning, et al.
0

We introduce a very weak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability, obtained by a non-standard definition of the trial space. As a consequence, the numerical approximation error is equal to the residual, which is particularly useful for a posteriori error estimation. For the arising discrete linear systems in space and time, we introduce efficient numerical solvers that appropriately exploit the equation structure, either at the preconditioning level or in the approximation phase by using a tailored Galerkin projection. This Galerkin method shows competitive behavior concerning wall-clock time, accuracy and memory as compared with a standard time-stepping method in particular in low regularity cases. Numerical experiments with a 3D (in space) wave equation illustrate our findings.

READ FULL TEXT

page 1

page 2

page 3

page 4

06/08/2021

A space-time Trefftz discontinuous Galerkin method for the linear Schrödinger equation

A space-time Trefftz discontinuous Galerkin method for the Schrödinger e...
03/07/2021

Numerical results for an unconditionally stable space-time finite element method for the wave equation

In this work, we introduce a new space-time variational formulation of t...
02/10/2021

Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

Based on relative energy estimates, we study the stability of solutions ...
08/22/2019

Galerkin-collocation approximation in time for the wave equation and its post-processing

We introduce and analyze a class of Galerkin-collocation discretization ...
08/12/2021

Feature Engineering with Regularity Structures

We investigate the use of models from the theory of regularity structure...
01/15/2022

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

We discuss two approaches for the formulation and implementation of spac...
03/31/2021

On the space-time discretization of variational retarded potential boundary integral equations

This paper discusses the practical development of space-time boundary el...