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Mixed methods and lower eigenvalue bounds

Dietmar Gallistl

TL;DR

The paper develops a general dual mixed finite-element framework to obtain computable lower bounds for the eigenvalues of symmetric positive definite PDE operators. A fundamental commutation property between continuous and discrete gradient-like operators, together with kernel inclusion and projection compatibility, yields a universal lower bound: the true eigenvalue is bounded below by the discrete value divided by a factor depending on a compact-embedding-derived constant. The approach applies to the Laplacian, general second-order scalar elliptic operators, linear elasticity (Lamé) eigenvalues, and the Steklov problem, providing explicit bounds in terms of mesh parameters and operator constants. While the method is low-order and relies on specific stability constants, it offers a unified mechanism for guaranteed bounds across a broad class of problems and can guide reliable spectral approximations in practice.

Abstract

It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility condition (inclusion of kernels) of the mixed scheme and on local constants related to compact embeddings, which are often known explicitly. Applications include scalar second-order elliptic operators, linear elasticity, and the Steklov eigenvalue problem.

Mixed methods and lower eigenvalue bounds

TL;DR

The paper develops a general dual mixed finite-element framework to obtain computable lower bounds for the eigenvalues of symmetric positive definite PDE operators. A fundamental commutation property between continuous and discrete gradient-like operators, together with kernel inclusion and projection compatibility, yields a universal lower bound: the true eigenvalue is bounded below by the discrete value divided by a factor depending on a compact-embedding-derived constant. The approach applies to the Laplacian, general second-order scalar elliptic operators, linear elasticity (Lamé) eigenvalues, and the Steklov problem, providing explicit bounds in terms of mesh parameters and operator constants. While the method is low-order and relies on specific stability constants, it offers a unified mechanism for guaranteed bounds across a broad class of problems and can guide reliable spectral approximations in practice.

Abstract

It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility condition (inclusion of kernels) of the mixed scheme and on local constants related to compact embeddings, which are often known explicitly. Applications include scalar second-order elliptic operators, linear elasticity, and the Steklov eigenvalue problem.
Paper Structure (8 sections, 11 theorems, 89 equations, 3 figures, 4 tables)

This paper contains 8 sections, 11 theorems, 89 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A basic commutation property
  3. Approximation of the eigenvalue problem
  4. Application to the Laplacian
  5. Scalar elliptic operator
  6. Application to linear elasticity
  7. Application to the Steklov eigenvalue problem
  8. Conclusive remarks

Key Result

Lemma 2.1

Let $(\cdot,\cdot)_U$ denote the inner product of $U$. The space $U_0$ with the inner product is a Hilbert space. If the inf-sup condition with some positive number $\beta>0$ is satisfied, the norms $\|G\cdot\|_a$ and $(\|\cdot\|_U^2+\|G\cdot\|_a^2)^{1/2}$ are equivalent on $U_0$.

Figures (3)

  • Figure 1: Initial triangulation of the L-shaped domain.
  • Figure 2: Coefficients $A$ (left) and $\gamma$ (middle) and the initial triangulation (right) in Example \ref{['ex:scalarellipt']}.
  • Figure 3: Cook's membrane with initial trianulation.

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Remark 3.3
  • Theorem 3.4: abstract lower bound
  • proof
  • ...and 21 more