Arbitrary high-order unconditionally stable methods for reaction-diffusion equations via Deferred Correction: Case of the implicit midpoint rule

In this paper we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. The IBVP is first discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order 2p+2 of accuracy at the stage p=0,1,2,... of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage p of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVP. This fully discretized scheme is unconditionlly stable with order 2p+2 of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular while an increment of one order is proven for shape-regular and quasi-uniform family of meshes. A numerical test with a bistable reaction-diffusion equation having a strong stiffness ratio is performed and shows that the orders 2,4,6,8 and 10 of accuracy in time are achieved with a very strong stability.


page 1

page 2

page 3

page 4


Mixed Finite Element Method and numerical analysis of a convection-diffusion-reaction model in a porous medium

A hydrogeological model for the spread of pollution in an aquifer is con...

An arbitrary order scheme on generic meshes for miscible displacements in porous media

We design, analyse and implement an arbitrary order scheme applicable to...

Uniform auxiliary space preconditioning for HDG methods for elliptic operators with a parameter dependent low order term

The auxiliary space preconditioning (ASP) technique is applied to the HD...

A projection hybrid high order finite volume/finite element method for incompressible turbulent flows

In this paper the projection hybrid FV/FE method presented in Busto et a...

Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model

In the present work, we investigate a model of the invasion of healthy t...

Re-initialization Free Level Set Evolution via Reaction Diffusion

This paper presents a novel reaction-diffusion (RD) method for implicit ...

A Conservative Finite Element ALE Scheme for Mass-Conserving Reaction-Diffusion Equations on Evolving Two-Dimensional Domains

Mass-conservative reaction-diffusion systems have recently been proposed...

Please sign up or login with your details

Forgot password? Click here to reset