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
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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.
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