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Optimal Local Error Estimates for Finite Element Methods with Measure-Valued Sources

Huadong Gao, Yuhui Huang

Abstract

We study finite element approximations of second-order elliptic problems with measure-valued right-hand sides supported on lower-dimensional sets. The exact solution generally lacks $H^1$-regularity due to the source singularity, which limits global convergence rates of numerical methods. Using a very weak solution framework, we establish well-posedness and global error estimates for standard Lagrange finite element methods on Lipschitz polyhedral/polygonal domains. By using interior estimates techniques, we prove optimal local $L^2$- and $H^1$-error estimates in subdomains that are strictly separated from the support of the measure. Extensive numerical experiments are provided to verify the theoretical results. These results show that for Lagrange FEMs solving elliptic problems with singular right-hand sides, the loss of global convergence is purely local, and that optimal convergence rates still hold away from the singular source.

Optimal Local Error Estimates for Finite Element Methods with Measure-Valued Sources

Abstract

We study finite element approximations of second-order elliptic problems with measure-valued right-hand sides supported on lower-dimensional sets. The exact solution generally lacks -regularity due to the source singularity, which limits global convergence rates of numerical methods. Using a very weak solution framework, we establish well-posedness and global error estimates for standard Lagrange finite element methods on Lipschitz polyhedral/polygonal domains. By using interior estimates techniques, we prove optimal local - and -error estimates in subdomains that are strictly separated from the support of the measure. Extensive numerical experiments are provided to verify the theoretical results. These results show that for Lagrange FEMs solving elliptic problems with singular right-hand sides, the loss of global convergence is purely local, and that optimal convergence rates still hold away from the singular source.
Paper Structure (8 sections, 10 theorems, 69 equations, 2 figures, 5 tables)

This paper contains 8 sections, 10 theorems, 69 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite element approximation and global estimates
  4. The very weak solution and a FEM
  5. Global estimates on general Lipschitz polygonal/polyhedral domains
  6. An improved local error estimate on subdomains away from source
  7. Numerical Experiments
  8. Conclusion

Key Result

Lemma 2.1

Let $f\in L^2(\Omega)$, the coefficient matrix $A \in W^{1,\infty}(\Omega)^{d \times d}$ be symmetric and uniformly elliptic, then the solution $u$ to the following problem satisfies where $\lambda_{\textrm{2D}}=\frac{\pi}{\max_j \Theta_j} > \frac{1}{2}$ with $\{\Theta_j\}$ denoting the re-entrant interior angles of $\Omega$, and $\lambda_{\textrm{3D}} > \frac{1}{2}$ which depends on both edges

Figures (2)

  • Figure 1: The domain (left); initial mesh (middle); numerical solution on a fine mesh using linear FEM (right). (Example \ref{['example:hexagon-domain']})
  • Figure 2: Isosurface plot of the numerical solution obtained with linear elements ($k=1$) (left); detailed view near the sources (right). (Example \ref{['example-3D']})

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2: Regularity of the solution with an interior source
  • Lemma 2.3: Interior regularity away from the source
  • Lemma 2.4: Local $L^2$ and energy estimate
  • Lemma 2.5: Local maximum norm estimate
  • Lemma 3.1: Well-posedness
  • Theorem 3.2
  • Theorem 3.3: Global error estimate
  • Remark 3.4
  • Theorem 4.1
  • ...and 4 more