Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations

by   Weizhang Huang, et al.

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.



There are no comments yet.


page 1

page 2

page 3

page 4


A High order Conservative Flux Optimization Finite Element Method for Diffusion Equations

This article presents a high order conservative flux optimization (CFO) ...

A High-Order Lower-Triangular Pseudo-Mass Matrix for Explicit Time Advancement of hp Triangular Finite Element Methods

Explicit time advancement for continuous finite elements requires the in...

Interplay between diffusion anisotropy and mesh skewness in Hybrid High-Order schemes

We explore the effects of mesh skewness on the accuracy of standard Hybr...

Krylov Subspace Recycling for Evolving Structures

Krylov subspace recycling is a powerful tool for solving long series of ...

On the smallest eigenvalue of finite element equations with meshes without regularity assumptions

A lower bound is provided for the smallest eigenvalue of finite element ...

Simple finite elements and multigrid for efficient mass-consistent wind downscaling in a coupled fire-atmosphere model

We present a simple finite element formulation of mass-consistent approx...

Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

We study continuous finite element dicretizations for one dimensional hy...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.