An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation

Boundary Value Problems, Jan 2017

An optimized splitting positive definite mixed finite element (SPDMFE) extrapolation approach based on proper orthogonal decomposition (POD) technique is developed for the two-dimension viscoelastic wave equation (2DVWE). The errors of the optimized SPDMFE extrapolation solutions are analyzed. The implement procedure for the optimized SPDMFE extrapolation approach is offered. Some numerical simulations have verified that the numerical conclusions are accordant with theoretical ones. This implies that the optimized SPDMFE extrapolation approach is viable and valid for solving 2DVWE. MSC: 74S10, 65M15, 35Q35.

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An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation

Luo and Teng Boundary Value Problems An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation Zhendong Luo 0 Fei Teng 1 0 School of Mathematics and Physics, North China Electric Power University , No. 2, Bei Nong Road, Changping District, Beijing, 102206 , China 1 School of Control and Computer Engineering, North China Electric Power University , No. 2, Bei Nong Road, Changping District, Beijing, 102206 , China An optimized splitting positive definite mixed finite element (SPDMFE) extrapolation approach based on proper orthogonal decomposition (POD) technique is developed for the two-dimension viscoelastic wave equation (2DVWE). The errors of the optimized SPDMFE extrapolation solutions are analyzed. The implement procedure for the optimized SPDMFE extrapolation approach is offered. Some numerical simulations have verified that the numerical conclusions are accordant with theoretical ones. This implies that the optimized SPDMFE extrapolation approach is viable and valid for solving 2DVWE. optimized splitting positive definite mixed finite element extrapolation approach; proper orthogonal decomposition technique; viscoelastic wave equation; error estimate; numerical simulation 1 Introduction In this article, we study the following two-dimensional viscoelastic wave equation Problem I For  < t < T , find u that satisfies ⎨⎪⎧ uut(tx–,yε,t) u=tψ–(γx, y,ut)=, f , onin∂ ,, all given functions, and T is the final time. For convenience and without losing universality, The main motivation and physical background of DVWE () are the modeling of the wave propagation and vibration phenomena in the viscoelastic matter (see, e.g., [, ]). Although there have been several numerical methods for DVWE (see, e.g., [–]), the splitting positive definite mixed finite element (SPDMFE) approach in [] is one of most novel © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ones for dealing with DVWE because it cannot only keep away from the restriction of the Brezzi-Babuška inequality and simultaneously find an unknown function (displacement) and its gradient (stress), but it can also ensure that the full discrete SPDMFE formulation is positive definite and robust. Reference [] has established a new SPDMFE formulation that includes fewer degrees of freedom than those and is different from that in [], but it still includes lots of degrees of freedom. Hence, a major key issue is how to lessen the degrees of freedom for the new SPDMFE formulation in [] so as to reduce the calculating load and the operation time in the numerical computation as well as obtain a desired accurate SPDMFE solution. Many reports have proven that the proper orthogonal decomposition (POD) technique is one of the most valid approaches lessening the degrees of freedom (i.e., unknowns) of numerical models for the time-dependent PDEs and alleviating the truncated error accumulation in the calculating course (see [–]). In fact, the POD technique offers an orthogonal basis to the given data, i.e., offers an optimal low order approximation to the given data. Though some optimized numerical formulations based on the POD technique for the time-dependent PDEs were presented (see [–]), these optimized formulations utilize all classical numerical solutions on the whole time interval [, T ] to formulate the POD bases and the optimized models, before recomputing the solutions on the same time interval [, T ], which actually belongs to the repeated calculations on the same time interval [, T ]. In order to eliminate those unrewarding repeated computations in the reduced-order finite element (FE) methods based on the POD technique, several reduced-order extrapolation FE methods based on the POD technique for hyperbolic equations, Sobolev equations, and the non-stationary parabolized Navier-Stokes equations have successfully been proposed by Luo et al. since  (see [–]). Nevertheless, as far as we know, there is not any article treating that the optimized SPDMFE extrapolation approach based on the POD technique for DVWE is set up or the implement procedure for the optimized SPDMFE extrapolation approach is offered. Therefore, in this article, we set up the optimized SPDMFE extrapolation approach based on the POD technique for DVWE and offer the error estimates for the optimized SPDMFE extrapolation solutions and the implement procedure for the optimized SPDMFE extrapolation approach. We adopt some numerical simulations to verify that the optimized SPDMFE extrapolation approach is viable and valid for dealing DVWE, too. The rest of the (...truncated)


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Zhendong Luo, Fei Teng. An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation, Boundary Value Problems, 2017, pp. 6, 2017, DOI: 10.1186/s13661-016-0739-8