A generalized inf-sup stable variational formulation for the wave equation

TL;DR

This paper introduces a generalized variational formulation for the wave equation that ensures unique solvability even when the right-hand side is in the dual space, enhancing the stability and analysis of numerical methods.

Contribution

It extends the classical variational formulation to include initial data information, enabling unique solutions for broader right-hand side spaces and improving numerical analysis frameworks.

Findings

Proves unique solvability for dual space right-hand sides.

Extends the ansatz space to incorporate initial condition information.

Facilitates the development of unconditionally stable space-time finite element methods.

Abstract

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for $f$ being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time $t=0$. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space-time finite element methods, and for the numerical analysis of boundary element methods to overcome the well-known norm gap in the analysis of boundary integral operators.