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A space-time adaptive boundary element method for the wave equation

Alessandra Aimi, Giulia Di Credico, Heiko Gimperlein, Chiara Guardasoni

TL;DR

We present a fully space-time adaptive boundary element method for the time-domain wave equation formulated as a boundary integral equation. The method employs residual-based a posteriori error indicators to drive local space-time refinements with tensor-product elements, integrated in a SOLVE–ESTIMATE–MARK–REFINE loop, and provides an algebraic reformulation suitable for updating the discrete system on adaptive meshes. Numerical experiments in 2D demonstrate improved convergence in the energy norm and reductions in degrees of freedom and memory for problems with spatial, temporal, or traveling singularities, while showing that the heuristic indicator tracks the energy error closely and the theoretical indicator estimates a weaker norm. The work lays groundwork for extensions to 3D, anisotropic refinements, hp methods, and nonlinear problems.

Abstract

This article initiates the study of space-time adaptive mesh refinements for time-dependent boundary element formulations of wave equations. Based on error indicators of residual type, we formulate an adaptive boundary element procedure for acoustic soft-scattering problems with local tensor-product refinements of the space-time mesh. We discuss the algorithmic challenges and investigate the proposed method in numerical experiments. In particular, we study the performance and improved convergence rates with respect to the energy norm for problems dominated by spatial, temporal or traveling singularities of the solution. The efficiency of the considered rigorous and heuristic a posteriori error indicators is discussed.

A space-time adaptive boundary element method for the wave equation

TL;DR

We present a fully space-time adaptive boundary element method for the time-domain wave equation formulated as a boundary integral equation. The method employs residual-based a posteriori error indicators to drive local space-time refinements with tensor-product elements, integrated in a SOLVE–ESTIMATE–MARK–REFINE loop, and provides an algebraic reformulation suitable for updating the discrete system on adaptive meshes. Numerical experiments in 2D demonstrate improved convergence in the energy norm and reductions in degrees of freedom and memory for problems with spatial, temporal, or traveling singularities, while showing that the heuristic indicator tracks the energy error closely and the theoretical indicator estimates a weaker norm. The work lays groundwork for extensions to 3D, anisotropic refinements, hp methods, and nonlinear problems.

Abstract

This article initiates the study of space-time adaptive mesh refinements for time-dependent boundary element formulations of wave equations. Based on error indicators of residual type, we formulate an adaptive boundary element procedure for acoustic soft-scattering problems with local tensor-product refinements of the space-time mesh. We discuss the algorithmic challenges and investigate the proposed method in numerical experiments. In particular, we study the performance and improved convergence rates with respect to the energy norm for problems dominated by spatial, temporal or traveling singularities of the solution. The efficiency of the considered rigorous and heuristic a posteriori error indicators is discussed.

Paper Structure

This paper contains 13 sections, 4 theorems, 39 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Weak formulation of the model problem
  3. Discretization
  4. Algorithmic details
  5. Space-time adaptive algorithm
  6. Algebraic reformulation of the discrete problem
  7. Implementation of error indicators
  8. Numerical results
  9. Example with smooth benchmark solution
  10. Example with peak-shaped datum
  11. Example with solution singular at the endpoints
  12. Example with solution singular in space and in time
  13. Conclusions

Key Result

Theorem 2.1

Let $r \in \mathbb{R}$.

Figures (20)

  • Figure 1: Starting mesh $\mathcal{T}_0$.
  • Figure 2: Example of mesh refinement.
  • Figure 3: Example \ref{['example1']}: exact solution (left) and refined mesh (right) using the theoretical error indicator and $\Theta=0.2$.
  • Figure 4: Example \ref{['example2']}: on the left, peak-shaped Dirichlet datum (left), numerical solution $\psi_{\Delta t, \Delta x}$ obtained with a uniform mesh of $N_x=N_t=160$ space-time elements (right).
  • Figure 5: Example \ref{['example2']}: decay of squared $L^2$ and energy errors w.r.t. degrees of freedom $DoF\!s=N_xN_t$.
  • ...and 15 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Theorem 3.1