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Solvers for mixed finite element problems using Poincaré operators based on spanning trees

Wietse M. Boon

Abstract

We propose an explicit construction of Poincaré operators for the lowest order finite element spaces, by employing spanning trees in the grid. In turn, a stable decomposition of the discrete spaces is derived that leads to an efficient numerical solver for the Hodge-Laplace problem. The solver consists of solving four smaller, symmetric positive definite systems. We moreover place the decomposition in the framework of auxiliary space preconditioning and propose robust preconditioners for elliptic mixed finite element problems. The efficiency of the approach is validated by numerical experiments.

Solvers for mixed finite element problems using Poincaré operators based on spanning trees

Abstract

We propose an explicit construction of Poincaré operators for the lowest order finite element spaces, by employing spanning trees in the grid. In turn, a stable decomposition of the discrete spaces is derived that leads to an efficient numerical solver for the Hodge-Laplace problem. The solver consists of solving four smaller, symmetric positive definite systems. We moreover place the decomposition in the framework of auxiliary space preconditioning and propose robust preconditioners for elliptic mixed finite element problems. The efficiency of the approach is validated by numerical experiments.

Paper Structure

This paper contains 13 sections, 14 theorems, 45 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminary definitions and notation conventions
  3. A Poincaré operator based on spanning trees
  4. A p-permitting decomposition
  5. The Poincaré operator
  6. Poincaré constants
  7. A solver for the Hodge-Laplace problem
  8. Auxiliary space preconditioning for elliptic problems
  9. Numerical results
  10. Solving the Hodge-Laplace problem
  11. Computing the Poincaré constant
  12. Preconditioning the projection problem
  13. Concluding remarks

Key Result

Lemma 2.3

The decomposition from ex: spanning trees is $p$-permitting (def: permitting).

Figures (1)

  • Figure 1: Decomposition of the facet space $P \Lambda^1$ on an unstructured grid in 2D. (Left) A spanning tree $\mathcal{T}_1^*$ connects all cells, including a fictitious "outside" cell through the domain boundary. The subspace $\bar{P} \Lambda^1$ is spanned by the basis functions associated with the facets crossed by $\mathcal{T}_1^*$. (Right) The complementary set of facets, illustrated in purple, forms a spanning tree $\mathcal{T}_1$ for the nodes. $\mathring{P} \Lambda^1$ is the subspace of $P \Lambda^1$ associated with these facets.

Theorems & Definitions (42)

  • Definition 1.1
  • Example 1.2
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4: Hodge decomposition
  • Example 2.5: Virtual element methods
  • Definition 2.6
  • Theorem 2.7
  • ...and 32 more