An efficient symmetric finite volume element method for second-order variable coefficient parabolic integro-differential
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An efficient symmetric finite volume element method for second-order variable coefficient parabolic integro-differential equations Xiaoting Gan1,2 · Dengguo Xu3 Received: 26 September 2019 / Revised: 4 August 2020 / Accepted: 27 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper is devoted to develop a symmetric finite volume element (FVE) method to solve second-order variable coefficient parabolic integro-differential equations, arising in modeling of nonlocal reactive flows in porous media. Based on barycenter dual mesh, one semi-discrete and two fully discrete backward Euler and Crank–Nicolson symmetric FVE schemes are presented. Then, the optimal order error estimates in L 2 -norm are derived for the semidiscrete and two fully discrete schemes. Numerical experiments are performed to examine the convergence rate and verify the effectiveness and usefulness of the new numerical schemes. Keywords Parabolic integro-differential equations · Barycenter dual mesh · Symmetric FVE schemes · L 2 -norm error estimates Mathematics Subject Classification 65N08 · 65N12 · 65N30
Communicated by Frederic Valentin. This work was supported by the National Natural Science Foundation of China (No. 61463002), the Science and Technology Program of Yunnan Province of China (2019FH001-079), and the Yunnan Provincial Department of Education Science Research Fund Project (No. 2019J0396).
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Xiaoting Gan [email protected] Dengguo Xu [email protected]
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School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
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School of Mathematics and Statistics, Chuxiong Normal University, Chuxiong 675000, People’s Republic of China
3
School of Automation, Beijing Institute of Technology, Beijing 100081, People’s Republic of China 0123456789().: V,-vol
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X. Gan, D. Xu
1 Introduction In this paper, we discuss and analyze a symmetric FVE method to the following class of second-order variable coefficient parabolic integro-differential initial boundary value problem: find u = u(x, t), such that: t ⎧ ∂u ⎪ ⎪ ∇ · (B∇u(s))ds = f (x, t), (x, t) ∈ × (0, T ], − ∇ · (A∇u) − ⎨ ∂t 0 (1.1) u(x, t) = 0, (x, t) ∈ ∂ × (0, T ], ⎪ ⎪ ⎩ x ∈ , u(x, 0) = u 0 (x), where ⊂ R 2 is a convex bounded polygonal domain with boundary ∂, A = {ai, j (x, t)} is a 2 × 2 real-valued symmetric and uniformly positive definite matrix, and B = {bi, j (x, t, s)} is a 2 × 2 matrix, x = (x, y). The nonhomogeneous term f (x, t) and u 0 (x) are known functions, which are assumed to be smooth, so that problem (1.1) has a unique solution in a certain Sobolev space. The problem of (1.1) is very important arises naturally in many applications, such as nonlocal reactive flows in porous media (Cushman 1993; Cushman et al. 1995), heat conduction in materials with memory (Renardy et al. (1987)), and nonFickian flow of fluid in porous media (Ewing et al. 2001). One important characteristic of these models is that they all expr
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