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Space-time virtual elements for the heat equation

Sergio Gómez, Lorenzo Mascotto, Andrea Moiola, Ilaria Perugia

TL;DR

The paper develops a space-time virtual element method for the heat equation on a space-time cylinder $Q_T$, using local VE spaces that solve heat problems with polynomial data and nonconforming global spaces linked by upwind projections. It establishes well-posedness through a VE Newton potential and a discrete inf-sup condition, and proves optimal a priori error estimates under isotropic mesh assumptions. Numerical results in $(1+1)$ and $(2+1)$ dimensions validate the theoretical rates, show robustness to incompatible data, and demonstrate high-order convergence with increasing degree $p$ or hp-adaptivity potential. The method offers a flexible, high-order framework on general prismatic space-time meshes, enabling time-slab decomposition, potential Trefftz variants, and adaptive refinement without dimension-dependent constraints.

Abstract

We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov-Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming in space, so that the analysis and the design of the method are independent of the spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. We prove well posedness and optimal error estimates for the scheme, and validate them with several numerical tests.

Space-time virtual elements for the heat equation

TL;DR

The paper develops a space-time virtual element method for the heat equation on a space-time cylinder , using local VE spaces that solve heat problems with polynomial data and nonconforming global spaces linked by upwind projections. It establishes well-posedness through a VE Newton potential and a discrete inf-sup condition, and proves optimal a priori error estimates under isotropic mesh assumptions. Numerical results in and dimensions validate the theoretical rates, show robustness to incompatible data, and demonstrate high-order convergence with increasing degree or hp-adaptivity potential. The method offers a flexible, high-order framework on general prismatic space-time meshes, enabling time-slab decomposition, potential Trefftz variants, and adaptive refinement without dimension-dependent constraints.

Abstract

We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov-Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming in space, so that the analysis and the design of the method are independent of the spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. We prove well posedness and optimal error estimates for the scheme, and validate them with several numerical tests.
Paper Structure (27 sections, 17 theorems, 120 equations, 8 figures)

This paper contains 27 sections, 17 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The model problem and its weak formulation
  3. Mesh assumptions
  4. The virtual element method
  5. Local virtual element spaces
  6. Polynomial projections
  7. Global virtual element spaces
  8. Discrete bilinear forms
  9. An admissible stabilization
  10. The method
  11. A glimpse on more general meshes
  12. Well posedness of the virtual element method
  13. A virtual element Newton potential
  14. A discrete inf-sup condition and well posedness of the method
  15. Convergence analysis
  16. ...and 12 more sections

Key Result

Proposition 1.1

If $f$ belongs to $L^2(0,T; H^{-1}(\Omega))$, then the variational formulation continuous-weak is well posed with the a priori bound

Figures (8)

  • Figure 1: (a) Left panel: a mesh satisfying assumption (G1). (b) Central panel: a mesh with nonmatching time-like facets. (c) Right panel: a mesh with nonmatching space-like facets.
  • Figure 2: $h$-dependence of the errors in \ref{['exact-errors']} for the patch tests with solution $u_p$ in \ref{['exact-patch-1D']}.
  • Figure 3: $h$-convergence of the errors in \ref{['exact-errors']} for the test case with smooth solution \ref{['smooth-test-case']}. The numbers in the yellow rectangles denote the experimental orders of convergence.
  • Figure 4: $h$-convergence of the errors in \ref{['exact-errors']} for the test case with
  • Figure 5: $h$-convergence of the errors in \ref{['exact-errors']} for the test case with singular solution $u_\alpha$\ref{['singular-solution']} with $\alpha = 0.75$.
  • ...and 3 more figures

Theorems & Definitions (37)

  • Proposition 1.1
  • Remark 1: Inhomogeneous initial and boundary conditions
  • Remark 2
  • Remark 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 27 more