Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems

Qiang Du, James M. Scott, Xiaochuan Tian

TL;DR

This work develops asymptotically compatible discretizations for parametrized nonlinear variational problems by extending Gamma-convergence based AC theory to nonlinear settings. It defines energy functionals $\\mathcal{E}_\\sigma$ on parametrized spaces $\\mathcal{T}_\\sigma$, proves $\\Gamma$-convergence of the extended energies $\\overline{\\mathcal{E}}_\\sigma$ to the local limit $\\overline{\\mathcal{E}}_\\infty$ as $\\sigma\to\\infty$, and shows convergence of Galerkin minimizers under convexity. It applies the framework to two nonlinear nonlocal models: (i) heterogeneous localization of the horizon with Neumann-type boundary conditions and $\\delta\to0$, and (ii) a domain varying with the horizon with Dirichlet nonlocal boundary conditions. Together, these results establish AC discretizations as robust, convergent tools bridging nonlocal and local theories and accommodating horizon- and domain- dependent parameter regimes.

Abstract

We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.

Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems

TL;DR

This work develops asymptotically compatible discretizations for parametrized nonlinear variational problems by extending Gamma-convergence based AC theory to nonlinear settings. It defines energy functionals on parametrized spaces , proves -convergence of the extended energies to the local limit as , and shows convergence of Galerkin minimizers under convexity. It applies the framework to two nonlinear nonlocal models: (i) heterogeneous localization of the horizon with Neumann-type boundary conditions and , and (ii) a domain varying with the horizon with Dirichlet nonlocal boundary conditions. Together, these results establish AC discretizations as robust, convergent tools bridging nonlocal and local theories and accommodating horizon- and domain- dependent parameter regimes.

Abstract

We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.
Paper Structure (6 sections, 8 theorems, 12 equations, 2 figures)

This paper contains 6 sections, 8 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Asymptotically compatible discretization
  3. Parametrized problems
  4. Finite dimensional approximations
  5. Gamma-convergence results
  6. Convergence of finite dimensional approximations to minimizers

Key Result

Theorem 1

Given assumption1, assumption2, and assumption3, there exists a $u_\sigma \in \mathcal{T}_\sigma$ satisfying eq:minprob for $\sigma \in (0,\infty)$. Furthermore, there exists a subsequence of $\{u_\sigma\}_\sigma$ (not relabeled) and there exists $u_\infty \in \mathcal{T}_\infty$ such that and $u_\infty$ satisfies eq:minprob with $\sigma=\infty$. If additionally $\mathcal{E}_\sigma$ is convex for

Figures (2)

  • Figure 1: A diagram of the gamma-convergence results for the functionals.
  • Figure 2: A diagram for asymptotically compatible schemes and convergence results.

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1: Convergence of minimizers as $\sigma\to\infty$
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 2: Convergence as $h \to 0$ with fixed $\sigma$
  • proof
  • Corollary 1
  • proof
  • ...and 9 more