A Splitting Mixed Covolume Method for Viscoelastic Wave Equations on Triangular Grids
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A Splitting Mixed Covolume Method for Viscoelastic Wave Equations on Triangular Grids Jie Zhao, Hong Li , Zhichao Fang, Yang Liu and Huifang Wang Abstract. A new splitting mixed covolume (SMCV) method is proposed for the viscoelastic wave equations by combining the splitting positive definite mixed finite element (SPDMFE) method with the mixed covolume (MCV) method. Based on the idea of the SPDMFE method, the difference between this method and the MCV method is that the proposed method does not need to solve the coupled system, thus reducing the scale of linear equations. By introducing a transfer operator, the semi-discrete and fully discrete SMCV schemes are proposed. The existence and uniqueness analysis of the semi-discrete scheme are given, and optimal a priori error estimates in different norm for these two schemes are derived. Finally, the feasibility and effectiveness of the proposed scheme are verified by some numerical results. Mathematics Subject Classification. 65M08, 65M15, 65M60. Keywords. Splitting mixed covolume method, viscoelastic wave equation, transfer operator, existence and uniqueness analysis, a priori error estimate.
1. Introduction We consider the following viscoelastic wave equations: ⎧ (a) c(x)utt − div(A(x)∇u + A(x)∇ut ) = f (x, t), x ∈ Ω, t ∈ J, ⎪ ⎨ ¯ (b) u(x, t) = 0, x ∈ ∂Ω, t ∈ J, (1.1) ⎪ ⎩ ¯ (c) u(x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω, This work was supported by National Natural Science Foundation of China (11701299, 11761053, 11661058), Natural Science Foundation of Inner Mongolia Autonomous Region (2016BS0105, 2017MS0107), Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07), and Prairie Talent Project of Inner Mongolia Autonomous Region.
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where Ω ⊂ R2 is a bounded convex polygonal region with a boundary ∂Ω, J = (0, T ] with 0 < T < ∞. The functions u0 (x), u1 (x), c(x) and f (x, t) are smooth enough. And there exist constants c0 and c1 to satisfy ¯ Moreover, we assume that A−1 (x) is Lipschitz 0 < c0 ≤ c−1 (x) ≤ c1 , ∀x ∈ Ω. continuous, and coefficient A(x) is a symmetric uniformly positive definite matrix function, that is, there exist positive constants α1 and α2 such that ¯ α1 ξ T ξ ≤ ξ T A(x)ξ ≤ α2 ξ T ξ, ∀ξ ∈ R2 , ∀x ∈ Ω. The viscoelastic wave equations are usually used to describe propagation phenomena when actual vibration passes through some viscoelastic medium. Many numerical methods have been studied to solve such problems. Larsson et al. [1] discussed the finite element (FE) method for the viscoelastic wave equation, and also considered the case of both smooth (and compatible) and less smooth data. Lin et al. [2] studied the semi-discrete FE approximation for the viscoelastic wave equation, and obtained the optimal error estimate by introducing Ritz–Volterra projection. Pani and Yuan [3] applied mixed finite element (MFE) method to solve the viscoelastic wave equation, and gave optimal error estimates in L2 -norm. Gao et al. [4] discussed the MFE m
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