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Localized Orthogonal Decomposition Method with $H^1$ Interpolation for Multiscale Elliptic Problem

Tao Yu, Xingye Yue

Abstract

This paper employs a localized orthogonal decomposition (LOD) method with $H^1$ interpolation for solving the multiscale elliptic problem. This method does not need any assumptions on scale separation. We give a priori error estimate for the proposed method. The theoretical results are conformed by various numerical experiments.

Localized Orthogonal Decomposition Method with $H^1$ Interpolation for Multiscale Elliptic Problem

Abstract

This paper employs a localized orthogonal decomposition (LOD) method with interpolation for solving the multiscale elliptic problem. This method does not need any assumptions on scale separation. We give a priori error estimate for the proposed method. The theoretical results are conformed by various numerical experiments.

Paper Structure

This paper contains 4 sections, 5 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. LOD Method with $H^1$ Interpolation for Multiscale Elliptic Problem
  3. Localization
  4. Numerical examples

Key Result

Lemma 2.1

There exists a positive constant $C$, which is independent on the mesh size $h$, such that for every interior nodal point $a\in K\in \mathcal{T}_h$ and $v\in V$, it holds that where $w_a \overset{\vartriangle}{=} \bigcup \{ K\in \mathcal{T}_h|a \in K\}$.

Figures (3)

  • Figure 1: Diffusion Coefficients: $A_1(x)$(left) and $A_2(x)$(right)
  • Figure 2: $L^2$ error (left) and $H^1$ error (right) for diffusion coefficient $A_1(x)$.
  • Figure 3: $L^2$ error (left) and $H^1$ error (right) for diffusion coefficient $A_2(x)$.

Theorems & Definitions (6)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Remark 3.2
  • Theorem 3.3