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Solving elliptic PDEs in unbounded domains

Doghonay Arjmand, Filip Marttala

Abstract

An accurate approximation of solutions to elliptic problems in infinite domains is challenging from a computational point of view. This is due to the need to replace the infinite domain with a sufficiently large and bounded computational domain, and posing artificial boundary conditions on the boundary of the truncated computational geometry, which will then pollute the solution in an interior region of interest. For elliptic problems with periodically varying coefficients (with a possibly unknown period), a modelling strategy based on exponentially regularized elliptic problem was previously developed and analysed. The main idea was to replace the infinite domain periodic problem with a regularized elliptic problem posed over a finite domain, while retaining an accuracy decaying exponentially with respect to the size of the truncated domain. In this article, we extend the analysis to problems, where no structural assumptions on the coefficient are assumed. Moreover, the analysis here uncovers an interesting property of the right hand side in the Fourier domain for the method to converge fast for problems beyond periodicity.

Solving elliptic PDEs in unbounded domains

Abstract

An accurate approximation of solutions to elliptic problems in infinite domains is challenging from a computational point of view. This is due to the need to replace the infinite domain with a sufficiently large and bounded computational domain, and posing artificial boundary conditions on the boundary of the truncated computational geometry, which will then pollute the solution in an interior region of interest. For elliptic problems with periodically varying coefficients (with a possibly unknown period), a modelling strategy based on exponentially regularized elliptic problem was previously developed and analysed. The main idea was to replace the infinite domain periodic problem with a regularized elliptic problem posed over a finite domain, while retaining an accuracy decaying exponentially with respect to the size of the truncated domain. In this article, we extend the analysis to problems, where no structural assumptions on the coefficient are assumed. Moreover, the analysis here uncovers an interesting property of the right hand side in the Fourier domain for the method to converge fast for problems beyond periodicity.

Paper Structure

This paper contains 11 sections, 18 theorems, 98 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence of elliptic equations with no regularization
  4. Improving the Decay by Regularization
  5. Decay Rates of the Modelling Error
  6. Decay of the Boundary Error
  7. Numerical Results
  8. Total Error
  9. Boundary Error
  10. Modelling Error
  11. Quasi-Periodic Coefficient

Key Result

Theorem 3

Let $u$ solve eqn_Main_Problem and $\Tilde{u}_R$ solve eq:finitedomainproblem with coefficients $a(\boldsymbol{x})\in\mathcal{M}(\alpha,\beta,\mathbb{R}^d)$ and $d\geq 3$. Let $L< \frac{R}{4}$ such that $K_L\subset K_{\frac{R}{4}}$. Further, let $\text{supp}(g(\boldsymbol{x}))=\Omega\subset K_L$ and where $C$ is a constant dependent only on $\beta$,$d$, and $\Omega$.

Figures (6)

  • Figure 1: Error convergence with respect to domain size $R$ for the 3D problem. The regularized method shows superior asymptotic behavior compared to the standard method
  • Figure 2: Error versus computational runtime for the 3D problem. The regularized method shows improved efficiency despite the higher computational cost for a given problem size
  • Figure 3: Relation between domain size $R$ and $L^2$-boundary-error for the 2D problem using the regularized method, demonstrating the exponential decay predicted by theory
  • Figure 4: Relation between domain size $R$ and $L^2$-boundary-error for the 2D problem using the standard method, showing first-order decay
  • Figure 5: Relation between domain size $R$ and $L^2$-modelling-error for the 2D problem, exhibiting exponential decay
  • ...and 1 more figures

Theorems & Definitions (40)

  • Theorem 3
  • Lemma 1
  • proof
  • Remark
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 30 more