Finite element methods for partial Volterra integro-differential equations on two-dimensional unbounded spatial domains

Abstract

The precise and effective way for solving a partial Volterra integro-differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain. Thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain, which is often called reduced problem. A feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit “nonlocal” properties. In this paper, the finite element method for the spatial space and the recently developed discontinuous Galerkin time-stepping method for the temporal space are considered to solve the reduced problem. The convergence rate, explicitly with those interesting parameters: spatial grid-size h, temporal mesh-size k, the location of the artificial boundary d, and the truncation number M, is obtained. This paper is the first theoretical analysis of the numerical methods for Volterra integro-differential equations on unbounded spatial domains.

Introduction

Let $\Omega \in {\mathbb{R}}^2$ be a semi-infinite strip domain with boundary Γ ≔ Γ_i ∪ Γ_U ∪ Γ_L (see Fig. 1). Consider the following initial-boundary-value problem for the two-dimensional reaction-diffusion equation with memory term:

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+{\int }_0^{t}k(x\text{,}t-\tau )u(x\text{,}\tau )\mathrm{d}\tau =\mathrm{\Delta }u+f(x\text{,}t)\text{,}x\in \Omega \text{,}t\in J≔[0\text{,}T]\text{,}$$

$${\left.u\right|}_{t=0}=u_0(x)\text{,}x\in \Omega \text{,}$$

$$u(x\text{,}t)=0\text{,}x\in \Gamma \text{,}t\in J\text{,}$$

$$u\to 0\text{as}|x|\to \infty \text{,}$$

where Δ denotes the Laplacian operator. Suppose supp{k(x, t) − k₀(t)}, supp{f} and supp{u₀} are compact. Furthermore assume f ∈ L²(Ω), u₀ ∈ H^1/2(Γ) and k₀ is continuous or weakly singular. For the end of computation we introduce an artificial boundary

$${\Gamma }_e≔\{(x_1\text{,}{x}_2):x_1=d\text{,}{x}_2\in [0\text{,}b]\}\text{,}$$

which divides the semi-infinite domain Ω into two parts (Ω_i and Ω_e), so that $\mathrm{supp}\{k(x\text{,}t)-k_0(t)\}\cup \mathrm{supp}\{f\}\cup \mathrm{supp}\{u_0\}\cup {\Gamma }_i\subset {\overline{\Omega }}_i$ (see also Fig. 1). Let d₀ be the minimum value of such d and define

$${\Gamma }_0≔\left\{(x_1\text{,}{x}_2):x_1=d_0\text{,}{x}_2\in [0\text{,}b]\right\}\text{.}$$

In the paper [11], the authors derive a boundary condition for the artificial boundary Γ_e and henceforth obtain a reduced problem on bounded domain Ω_i. The reduced problem with this kind of nonlocal artificial boundary condition is verified to be well-posed (see [11]). The finite element method and DG time-stepping method are applied to this reduced problem and the a priori error bound is derived. We remark that if k ≡ 0 or if k is the δ function, then the problem turns out to be the heat equation on unbounded domains. The papers [12], [13] derive the artificial boundary conditions (see also [10] and Table 1 in this paper for the summary of the connection). However, those papers are not able to derive the error bounds of the numerical methods applied. The history of artificial boundary methods for solving partial differential equations may go back to the works [14], [15], [16].

The discontinuous Galerkin (DG) method for ODEs and for time-stepping in parabolic PDEs was analyzed in details by Delfour et al. [8] and more recently by Schötzau and Schwab [20], [21] (see also for additional references). For Volterra integral equations the DG method was first studied by Shaw and Whiteman [22] in 1996 (compare also [23] and its list of references). In 1998 Larsson et al. [17] derived a priori error estimates of the DG method for linear parabolic integro-differential equations with weakly singular kernel. Recently an hp-version of the DG method for parabolic Volterra integro-differential equations with weakly singular kernels was investigated by Brunner and Schötzau [3]. The paper [19] gave the a posteriori error estimates of the DG method for non-standard Volterra integro-differential equations. See also Brunner [2] and Cockburn et al. [6] for the survey of this method and the comprehensive references.