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Using sparse grids and the \(L^2\) -orthogonality of prewavelets, one can apply a Ritz-Galerkin discretization of elliptic partial differential equations (PDEs) with variable coefficients even for dimension \(d>3\) . This leads to a linear equation system with \(O(N(\log (N)^{d-1}))\) unknowns. The theory of regular sparse grids can be extended to locally adaptive sparse grids. These are needed for several applications like PDEs with corner singularities or the high-dimensional Schrödinger equation. We introduce two different refinement strategies to obtain locally adaptive sparse grids. A numerical example shows how the correct choice of such a refinement strategy has major impact on the convergence rate of the discretization.

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Solving PDEs With Variable Coefficients on Locally Adaptive Sparse Grids and Corresponding Refinement Strategies

  • 会议论文

  • 发表日期：

    2025年

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  • Riccarda Scherner-Grießhammer,
  • Christoph Pflaum

摘要

Using sparse grids and the \(L^2\) -orthogonality of prewavelets, one can apply a Ritz-Galerkin discretization of elliptic partial differential equations (PDEs) with variable coefficients even for dimension \(d>3\) . This leads to a linear equation system with \(O(N(\log (N)^{d-1}))\) unknowns. The theory of regular sparse grids can be extended to locally adaptive sparse grids. These are needed for several applications like PDEs with corner singularities or the high-dimensional Schrödinger equation. We introduce two different refinement strategies to obtain locally adaptive sparse grids. A numerical example shows how the correct choice of such a refinement strategy has major impact on the convergence rate of the discretization.