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Spatial Approximation for Evolutionary Equations

Andreas Buchinger, Christian Seifert, Sascha Trostorff, Marcus Waurick

TL;DR

This work develops a general spatial approximation theory for autonomous evolutionary equations in Hilbert spaces, grounded in Picard's well-posedness framework. It introduces a convergence meta-theorem: consistency of spatial discretisations $(zM_n(z)+A_n)^{-1}$ to the continuous $(zM(z)+A)^{-1}$ together with stability guarantees yields convergence of the approximate solutions in weighted time-space spaces. The theory is demonstrated through a spectral-Galerkin discretisation of the heat equation, proving strong convergence of discrete resolvents and thus of the solutions. The results offer a unified, model-independent basis for spatial discretisation of evolutionary PDEs and have potential implications for homogenisation and domain-shape variability problems.

Abstract

We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the approximation result, we apply it to a spatial discretisation of the heat equation using spectral methods.

Spatial Approximation for Evolutionary Equations

TL;DR

This work develops a general spatial approximation theory for autonomous evolutionary equations in Hilbert spaces, grounded in Picard's well-posedness framework. It introduces a convergence meta-theorem: consistency of spatial discretisations to the continuous together with stability guarantees yields convergence of the approximate solutions in weighted time-space spaces. The theory is demonstrated through a spectral-Galerkin discretisation of the heat equation, proving strong convergence of discrete resolvents and thus of the solutions. The results offer a unified, model-independent basis for spatial discretisation of evolutionary PDEs and have potential implications for homogenisation and domain-shape variability problems.

Abstract

We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the approximation result, we apply it to a spatial discretisation of the heat equation using spectral methods.

Paper Structure

This paper contains 4 sections, 10 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Evolutionary Equations
  3. Spatial Approximation
  4. Example: The Heat Equation

Key Result

Lemma 2.1

Let $H$ be a Hilbert space, $\nu\in\mathbb{R}$, and $A\colon\mathop{\mathrm{dom}}\nolimits(A)\subseteq H\to H$ skew-selfadjoint. Regard $\mathop{\mathrm{dom}}\nolimits (A)$ as a Hilbert space endowed with the graph inner product induced by $A$. Then, is skew-selfadjoint.

Theorems & Definitions (33)

  • Lemma 2.1
  • Theorem 2.2: Picard's Well-Posedness Theorem
  • proof
  • Remark 3.1
  • Definition
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 23 more