Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical approximation of linear parabolic evolution equations revisited

Øyvind Stormark Auestad

Abstract

We obtain rates of convergence of numerical approximations of abstract linear parabolic evolution equations in Banach spaces. Our estimates extend known results from the literature of finite element approximations of parabolic equations to more general equations and numerical approximation methods. As an example, we consider parabolic equations on surfaces and surface finite element approximations.

Numerical approximation of linear parabolic evolution equations revisited

Abstract

We obtain rates of convergence of numerical approximations of abstract linear parabolic evolution equations in Banach spaces. Our estimates extend known results from the literature of finite element approximations of parabolic equations to more general equations and numerical approximation methods. As an example, we consider parabolic equations on surfaces and surface finite element approximations.
Paper Structure (9 sections, 39 theorems, 243 equations)

This paper contains 9 sections, 39 theorems, 243 equations.

Table of Contents

  1. introduction
  2. Semidiscrete and fully discerete error estimates
  3. Semidiscrete error estimate
  4. Fully discrete error estimate
  5. Semidiscrete and fully discrete error estimates in different norms, and for less regular data
  6. Example: surface finite element approximation of a parabolic equation
  7. Proofs of analytic semigroup properties
  8. Proofs related to rational approximation of analytic semigroups
  9. Differential operators on surfaces

Key Result

Theorem 2.2

Under Assumption assumption:abstract-model, there is $C, \epsilon > 0$ such that for $\theta \in [0,r]$ and $\rho \in [0,\theta]$.

Theorems & Definitions (79)

  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • proof : Proof of Theorem \ref{['theorem:1']}
  • ...and 69 more