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

Olaf Steinbach*, Marco Zank

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number125457
Number of pages24
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - Jan 2022


  • Bijectivity
  • Inf-sup stability condition
  • Variational formulation
  • Wave equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Fields of Expertise

  • Information, Communication & Computing

Cite this