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<p>In this paper, we present a conforming space–time discretization of the wave equation based on a first-order-in-time variational formulation. Our method extends the scheme of French and Peterson (1996), incorporating exponential weights in time, which yield an inf–sup stability condition for arbitrary choices of discrete subspaces, including spline spaces, without restrictions on the mesh size or time step. Moreover, using elliptic projections, we derive optimal convergence rates in both the energy and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norms for sufficiently smooth solutions and for any choice of space–time tensor product subspaces satisfying standard approximation assumptions. Numerical examples are provided to support the theoretical findings.</p>

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Inf–Sup Stable Space–Time Discretization of the Wave Equation Based on a First-Order-In-Time Variational Formulation

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  • 发表日期：

    2026年5月4日

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  • Matteo Ferrari,
  • Ilaria Perugia,
  • Enrico Zampa

摘要

In this paper, we present a conforming space–time discretization of the wave equation based on a first-order-in-time variational formulation. Our method extends the scheme of French and Peterson (1996), incorporating exponential weights in time, which yield an inf–sup stability condition for arbitrary choices of discrete subspaces, including spline spaces, without restrictions on the mesh size or time step. Moreover, using elliptic projections, we derive optimal convergence rates in both the energy and \(L^2\) L 2 norms for sufficiently smooth solutions and for any choice of space–time tensor product subspaces satisfying standard approximation assumptions. Numerical examples are provided to support the theoretical findings.