Least-squares Galerkin procedures for parabolic integro-differential equations

doi:10.1016/S0096-3003(03)00304-7

Applied Mathematics and Computation

Volume 150, Issue 3, 17 March 2004, Pages 749-762

https://doi.org/10.1016/S0096-3003(03)00304-7 Get rights and content

Abstract

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in $H(div;Ω)×H^{1} (Ω)$ and $(L^{2} (Ω))^{2} ×L^{2} (Ω)$ , respectively.

Introduction

We consider the following integro-differential equation of parabolic type: $c(x) ∂ u ∂ t − ∇ a(x) ∇ u+b(x)∫_{0}^{t} ∇ u(x,τ) d τ +qu=f(x,t), in Ω×J u(x,t)|_{Γ_{D}×J} =0, a(x) ∇ u+b(x)∫_{0}^{t} ∇ u(x,τ) d τ ·ν(x)|_{Γ_{N}×J} =0, u(x,0)=u_{0} (x),$ where J=(0,T) is the time interval, $Ω$ is a bounded polygonal domain in R² with a Lipschitz continuous boundary Γ. Γ=Γ_D∪Γ_N and ν is the unit vector normal to Γ_N. q=q(x)⩾0 and f=f(x,t) in (1.1) are given functions. We shall make the following assumptions on the coefficients c, a and b: there exist positive constants k₁, k₂ and $a_{∗}$ , $a^{∗}$ such that $0<k_{1} ⩽c(x)⩽k_{2}, 0<a_{∗} ⩽a(x)⩽a^{∗}, |b(x)|⩽k_{2} .$

Problem (1.1) can arise from many physical processes in which it is necessary to take into account the effects of memory due to the deficiency of the usual diffusion equations. For example, it can serve as a model in some gas diffusion problems [1] and in some heat transfer problems with memory [2]. For linear or nonlinear integro-differential equations, the standard finite element methods have received considerable attention and are well studied. See, for example, [3], [4], [5]. Recently, mixed methods to approximate such problems have been formulated in [6].

The disadvantage of the mixed element method is that the LBB condition must be satisfied by the finite element spaces. In this paper, we propose and analyze the least-squares Galerkin finite element method for parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition, so we can select the finite element spaces without constrains. We formulate two novel numerical methods and give the optimal convergent rates in $H(div;Ω)×H^{1} (Ω)$ and $(L^{2} (Ω))^{2} ×L^{2} (Ω)$ norms, respectively.

The paper is organized as follows. In Section 2 we formulate two least-squares Galerkin finite element procedures. The convergence theory on these algorithms is established in Section 3.

Throughout this paper, the symbols K and δ are used to denote a generic constant and a generic small positive constant, respectively. We also use the usual definitions and notations of Sobolev spaces as in [7].

Section snippets

Least-squares Galerkin finite element procedures

Let W^k,p(k⩾0,1⩽p⩽∞) be Sobolev spaces defined on $Ω$ with usual norms $∥·∥_{W^{k,p}(Ω)}$ and $H^{k} (Ω)=W^{k,2} (Ω)$ . Define inner products and spaces as follows. $(u,v)=∫_{Ω} uv d x ∀u,v∈L^{2} (Ω),$ $(σ,ω)=∑_{i=1}^{2} (σ_{i},ω_{i}) ∀σ=(σ_{1},σ_{2}), ω=(ω_{1},ω_{2})∈L^{2} (Ω)^{2} .$ $H={w∈L^{2} (Ω)^{2}; div w∈L^{2} (Ω),ω·ν=0 on Γ_{N}},$ $S={v∈H^{1} (Ω),v=0 on Γ_{D}}.$ Given a time step Δt=T/N, where N is a positive integer, we shall approximate the solution at times t_n=nΔt, n=0,1,…,N.

Let T_{h_σ} and T_{h_u} be two families of finite element partitions of the domain $Ω$ , which are identical or not, h_σ and h_u

Convergence analysis

In this section, we analyze the convergence of the two procedures. First we analyze Scheme I.

Combining , , we obtain $a_{N} ((σ^{n} −σ^{n}_{h},u^{n} −u^{n}_{h}),(ψ_{h},v_{h}))= 1 c (c(u^{n−1} −u^{n−1}_{h})+ Δ tR^{n}_{1}),cv_{h} + Δ t(div ψ_{h} +qv_{h}) + Δ t a(x)^{−1} R^{n}_{2} − Δ tb(x)∑_{i=0}^{n−1} ∇ (u^{i} −u^{i}_{h})(x),ψ_{h} +a(x) ∇ v_{h} ∀(ψ_{h},v_{h})∈H_{h_{σ}} ×S_{h_{u}} .$

Let θⁿ=Ruⁿ−uⁿ_h, ρⁿ=uⁿ−Ruⁿ, πⁿ=Qσⁿ−σⁿ_h, ϵⁿ=σⁿ−Qσⁿ. The estimates for ρⁿ and ϵⁿ were given in , , we need to estimate θⁿ and πⁿ. From (3.1) we see that (πⁿ,θⁿ) satisfies the following error equation: $1 c (c(θ^{n} + Δ t(div π^{n} +qθ^{n})),cv_{h})+ Δ t(div ψ_{h} +qv_{h}) + Δ t(a(x)^{−1}$

Acknowledgements

Supported by the National Natural Science Foundation of China grant no. 10071044 and the Research Fund for Doctoral Program of High Education by State Education Ministry of China.

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Least-squares Galerkin procedures for parabolic integro-differential equations

Abstract

Introduction

Section snippets

Least-squares Galerkin finite element procedures

Convergence analysis

Acknowledgements

On some nonlinear problems of diffusion

A general theory of heat conduction with finite wave speeds

Arch. Rational Mech. Anal.

A priori L2 error estimates for finite element methods for nonlinear diffusion equations with memory

SIAM J. Numer. Anal.

Non-classical H1 projection and Galerkin methods for nonlinear parabolic integro-differential equations

Calcolo

A priori L² error estimates for finite element methods for nonlinear diffusion equations with memory

Non-classical H¹ projection and Galerkin methods for nonlinear parabolic integro-differential equations