On the Algebraic Construction of Sparse Multilevel Approximations of Elliptic Tensor Product Problems

Abstract

We consider the solution of elliptic problems on the tensor product of two physical domains as for example present in the approximation of the solution covariance of elliptic partial differential equations with random input. Previous sparse approximation approaches used a geometrically constructed multilevel hierarchy. Instead, we construct this hierarchy for a given discretized problem by means of the algebraic multigrid method. Thereby, we are able to apply the sparse grid combination technique to problems given on complex geometries and for discretizations arising from unstructured grids, which was not feasible before. Numerical results show that our algebraic construction exhibits the same convergence behaviour as the geometric construction, while being applicable even in black-box type PDE solvers.

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Correspondence to Peter Zaspel.

Additional information

This work is funded by the Swiss National Science Foundation (SNF) under Project Number \(407540\_167186\).

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Harbrecht, H., Zaspel, P. On the Algebraic Construction of Sparse Multilevel Approximations of Elliptic Tensor Product Problems. J Sci Comput 78, 1272–1290 (2019). https://doi.org/10.1007/s10915-018-0807-6

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Keywords

  • Elliptic boundary value problem
  • Sparse tensor product approximation
  • Combination technique
  • Algebraic multigrid
  • Uncertainty quantification

Mathematics Subject Classification

  • 65N30
  • 65N22
  • 65N55
  • 65N50
  • 65F10
  • 65Y20
  • 65Y05