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Computational multiscale methods for nondivergence-form elliptic partial differential equations

Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler

TL;DR

Novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition are proposed.

Abstract

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.

Computational multiscale methods for nondivergence-form elliptic partial differential equations

TL;DR

Novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition are proposed.

Abstract

This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence-form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence-form.
Paper Structure (43 sections, 9 theorems, 92 equations, 10 figures)

This paper contains 43 sections, 9 theorems, 92 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem setting and well-posedness
  3. Framework
  4. Well-posedness
  5. Numerical homogenization scheme
  6. Fine-scale space
  7. Triangulation
  8. Quantities of interest and the space of fine-scale functions
  9. Connection to the Morley finite element space
  10. Projectors onto the fine-scale space
  11. Ideal numerical homogenization scheme
  12. $a$-orthogonal decompositions of $V$
  13. Ideal numerical homogenization
  14. Construction of a coarse-scale space
  15. Construction of a local basis
  16. ...and 28 more sections

Key Result

Theorem 2.1

The following assertions hold true. In particular, the problem u prob has a unique strong solution $u\in V$.

Figures (10)

  • Figure 1: Relative errors for the periodic problem with vanishing lower-order terms ($A = A^{(1)}$, $b = 0$, $c = 0$, $f = f^{(1)}$).
  • Figure 2: Relative errors for the periodic problem with non-vanishing lower-order terms ($A = A^{(1)}$, $b = b^{(1)}$, $c = c^{(1)}$, $f = f^{(1)}$) and $\lambda=1$.
  • Figure 3: Illustration of the coefficients chosen in \ref{['Subsec: Cr ex']}.
  • Figure 4: Relative errors for the crack problem with vanishing lower-order terms ($A = A^{(2)}$, $b = 0$, $c = 0$, $f = f^{(2)}$).
  • Figure 5: Relative errors for the crack problem with non-vanishing lower-order terms ($A = A^{(2)}$, $b = b^{(2)}$, $c = c^{(2)}$, $f = f^{(2)}$) and $\lambda=2$.
  • ...and 5 more figures

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2: Properties of $\gamma$
  • Theorem 2.1: Well-posedness
  • Lemma 2.1: Properties of the maps $a$ and $F$
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.1: Properties of $\tilde{U}_H$ and $\tilde{V}_H$
  • proof
  • Theorem 3.1: Analysis of the ideal discrete problem
  • proof
  • ...and 13 more