Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem

Timo Sprekeler

TL;DR

This work develops a rigorous homogenization theory for nondivergence-form elliptic PDEs with measurable, periodic coefficients satisfying the Cordes condition, proving convergence to a homogenized problem with an effective matrix $\bar{A}=\int_Y rA$ where $r$ is the invariant measure of $A$. It introduces a constructive, invariant-measure-based framework that avoids divergence-form reformulations and provides sharp corrector estimates up to order $\varepsilon^2$ under suitable regularity, including the notion of type-$\varepsilon^2$ diffusion matrices. A finite element method is designed to approximate the invariant measure and the homogenized problem, with rigorous error bounds linking $r_h$ to $\bar{A}_h$ and to the homogenized solution $u_h$. Numerical experiments illustrate the method in 2D with discontinuous coefficients, demonstrating convergence behavior and practical viability of the approach for challenging coefficient structures. Overall, the paper extends homogenization and numerical techniques to the setting of bounded, measurable, periodic coefficients under Cordes, enabling accurate computation of homogenized models in nondivergence contexts.

Abstract

We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $Ω\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic and symmetric diffusion matrix $A$ is merely assumed to be essentially bounded and (if $n>2$) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem, we obtain $L^2$-bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of Hölder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.

Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem

TL;DR

This work develops a rigorous homogenization theory for nondivergence-form elliptic PDEs with measurable, periodic coefficients satisfying the Cordes condition, proving convergence to a homogenized problem with an effective matrix where is the invariant measure of . It introduces a constructive, invariant-measure-based framework that avoids divergence-form reformulations and provides sharp corrector estimates up to order under suitable regularity, including the notion of type- diffusion matrices. A finite element method is designed to approximate the invariant measure and the homogenized problem, with rigorous error bounds linking to and to the homogenized solution . Numerical experiments illustrate the method in 2D with discontinuous coefficients, demonstrating convergence behavior and practical viability of the approach for challenging coefficient structures. Overall, the paper extends homogenization and numerical techniques to the setting of bounded, measurable, periodic coefficients under Cordes, enabling accurate computation of homogenized models in nondivergence contexts.

Abstract

We study the homogenization of the equation posed in a bounded convex domain subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic and symmetric diffusion matrix is merely assumed to be essentially bounded and (if ) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem, we obtain -bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of Hölder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.
Paper Structure (23 sections, 17 theorems, 68 equations, 2 figures)

This paper contains 23 sections, 17 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Homogenization
  4. Preliminaries
  5. Periodic problems and invariant measures
  6. The convergence result
  7. Corrector estimates
  8. Type-$\varepsilon^2$ diffusion matrices
  9. Numerical Methods
  10. Finite element approximation of the invariant measure
  11. Approximation of $\tilde{r}$ via $H^2_{\mathrm{per},0}(Y)$-conforming FEM for $\psi$
  12. Approximation of $\tilde{r}$ via $H^1_{\mathrm{per},0}(Y;\mathbb{R}^n)$-conforming FEM
  13. Approximation of $r$
  14. Approximation of the homogenized problem
  15. Numerical Experiments
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain. Let $A\in \mathcal{M}(\lambda,\Lambda)$, $f\in L^2(\Omega)$, $g\in H^2(\Omega)$, and suppose that Cordes holds. Then, for any $\varepsilon > 0$ there exists a unique solution $u_{\varepsilon}\in H^2(\Omega)$ to ueps problem, and we have the for some constant $C = C(\mathrm{diam}(\Omega),\lambda,\Lambda,n,\delta) > 0$.

Figures (2)

  • Figure 1: Plot of the function $a$ defined in \ref{['A for numexp']} and plot of the invariant measure $r$ to $A=\mathrm{diag}(1-a,a)$ given by \ref{['r fornumexp']}.
  • Figure 2: Approximation error for the approximation of the invariant measure $r$ and the effective coefficient matrix $\bar{A}$ from Section \ref{['Sec: Exp']}. We observe two curves, corresponding to whether or not there are elements of the triangulation whose interior intersect the line $\{y_1 = \frac{1}{2}\}$, i.e., the set along which $r$ exhibits a jump.

Theorems & Definitions (41)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: well-posedness and uniform $H^2$-bound
  • Lemma 3.1: the bilinear form $b_{\mu}$
  • Remark 3.1
  • Definition 3.1: invariant measure
  • Theorem 3.1: existence, uniqueness and properties of invariant measures
  • Theorem 3.2: analysis of the problem \ref{['q equation']}
  • Theorem 3.3: analysis of the problem \ref{['v equation']}
  • Remark 3.2
  • ...and 31 more