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Monotonicity of $Q^3$ spectral element method for discrete Laplacian

Logan J. Cross, Xiangxiong Zhang

TL;DR

For the $Q^3$ spectral element method for the two-dimensional Laplacian, it is proved its stiffness matrix is a product of four M-matrices thus it is monotone and can be regarded as a fifth order accurate finite difference scheme.

Abstract

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are fourth order accurate schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme.

Monotonicity of $Q^3$ spectral element method for discrete Laplacian

TL;DR

For the spectral element method for the two-dimensional Laplacian, it is proved its stiffness matrix is a product of four M-matrices thus it is monotone and can be regarded as a fifth order accurate finite difference scheme.

Abstract

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are fourth order accurate schemes, which is either an M-matrix or a product of two M-matrices. For the spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme.

Paper Structure

This paper contains 21 sections, 6 theorems, 39 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Monotone high order schemes
  3. Monotonicity of $Q^k$ spectral element method
  4. Contribution and organization of the paper
  5. Classical monotone high order finite difference schemes
  6. 9-point scheme
  7. The Bramble and Hubbard's scheme
  8. Monotone high order finite element methods on structured meshes
  9. Finite element method with the simplest quadrature
  10. The $P^2$ finite element method
  11. The $Q^2$ spectral element method
  12. The $Q^3$ spectral element method
  13. Lorenz's condition for monotonicity
  14. Monotonicity of $Q^3$ spectral element method on a uniform mesh
  15. The main approach for proving monotonicity
  16. ...and 6 more sections

Key Result

theorem 1

For a real square matrix $A$ with positive diagonal entries and non-positive off-diagonal entries, $A$ is a nonsingular M-matrix if all the row sums of $A$ are non-negative and at least one row sum is positive.

Figures (10)

  • Figure 1: An illustration of Lagrangian $P^2$ element and the simple third order accurate quadrature using vertices and edge centers.
  • Figure 2: An illustration of Lagrangian $Q^2$ element and the $3\times3$ Gauss-Lobatto quadrature.
  • Figure 3: The simple quadrature on two triangles give a quadrature on a square.
  • Figure 4: An illustration of a mesh for $Q^3$ element and the $4\times4$ Gauss-Lobatto quadrature.
  • Figure 5: Three adjacent 1D cells for $P^3$ elements using 4-point Gauss-Lobatto quadrature.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Remark 1
  • theorem 1
  • theorem 2
  • Remark 2
  • lemma thmcounterlemma
  • proof
  • Definition 1
  • theorem 3
  • theorem 4: Lorenz's condition
  • Proposition 1
  • ...and 1 more