Volume 56, Number 4, July-August 2022
|Page(s)||1173 - 1198|
|Published online||27 June 2022|
An ultraweak space-time variational formulation for the wave equation: Analysis and efficient numerical solution
Ulm University, Institute for Numerical Mathematics, Helmholtzstr. 18, 89081 Ulm, Germany
2 Università di Bologna, Centro AM 2, Dipartimento di Matematica, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
* Corresponding author: firstname.lastname@example.org
Accepted: 7 April 2022
We introduce an ultraweak 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.
Mathematics Subject Classification: 35L15 / 65M15 / 65M60
Key words: Wave equation / ultraweak formulation / tensorproduct / numerical solvers
© The authors. Published by EDP Sciences, SMAI 2022
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