Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate...

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Published inOpen mathematics (Warsaw, Poland) Vol. 22; no. 1; pp. 90 - 116
Main Author Wang, Keyan
Format Journal Article
LanguageEnglish
Published Warsaw De Gruyter 27.08.2024
De Gruyter Poland
Subjects
35L20
65M60
Algorithms
Approximation
Asymptotic methods
error estimate
expanded mixed finite element method
Finite element method
Grid method
hyperbolic equation
Mathematical analysis
Numerical stability
Two dimensional analysis
two-grid method
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Abstract In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size and the fine grid size satisfy ( ), where is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
AbstractList In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size HH and the fine grid size hh satisfy h=O(H(2k+1)⁄(k+1))h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) (k≥1k\ge 1), where kk is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H and the fine grid size h satisfy h=O(H(2k+1)⁄(k+1)) (k≥1), where k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size and the fine grid size satisfy ( ), where is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.
Author Wang, Keyan
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SubjectTerms 35L20
65M60
Algorithms
Approximation
Asymptotic methods
error estimate
expanded mixed finite element method
Finite element method
Grid method
hyperbolic equation
Mathematical analysis
Numerical stability
Two dimensional analysis
two-grid method
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Title Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
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