A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type
We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-ord...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2013/787891 |
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| author | Yang Liu Hong Li Zhichao Fang Siriguleng He Jinfeng Wang |
| author_facet | Yang Liu Hong Li Zhichao Fang Siriguleng He Jinfeng Wang |
| author_sort | Yang Liu |
| collection | DOAJ |
| description | We propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable (L2(Ω))2 space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution λ=∇u. For a priori error estimates based on both semidiscrete and fully
discrete schemes, we introduce a new expanded mixed projection and some important lemmas.
We derive the optimal a priori error estimates in L2 and H1-norm for both the scalar unknown u and the diffusion term γ and a priori error estimates in (L2)2-norm for its gradient λ and its flux σ (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method. |
| format | Article |
| id | doaj-art-c9613a917e1041e184acf8fc0602ff80 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-c9613a917e1041e184acf8fc0602ff802025-02-03T06:45:18ZengWileyAdvances in Mathematical Physics1687-91201687-91392013-01-01201310.1155/2013/787891787891A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic TypeYang Liu0Hong Li1Zhichao Fang2Siriguleng He3Jinfeng Wang4School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, ChinaSchool of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, ChinaWe propose and analyze a new numerical method, called a coupling method based on a new expanded mixed finite element (EMFE) and finite element (FE), for fourth-order partial differential equation of parabolic type. We first reduce the fourth-order parabolic equation to a coupled system of second-order equations and then solve a second-order equation by FE method and approximate the other one by a new EMFE method. We find that the new EMFE method’s gradient belongs to the simple square integrable (L2(Ω))2 space, which avoids the use of the classical H(div; Ω) space and reduces the regularity requirement on the gradient solution λ=∇u. For a priori error estimates based on both semidiscrete and fully discrete schemes, we introduce a new expanded mixed projection and some important lemmas. We derive the optimal a priori error estimates in L2 and H1-norm for both the scalar unknown u and the diffusion term γ and a priori error estimates in (L2)2-norm for its gradient λ and its flux σ (the coefficients times the negative gradient). Finally, we provide some numerical results to illustrate the efficiency of our method.http://dx.doi.org/10.1155/2013/787891 |
| spellingShingle | Yang Liu Hong Li Zhichao Fang Siriguleng He Jinfeng Wang A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type Advances in Mathematical Physics |
| title | A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type |
| title_full | A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type |
| title_fullStr | A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type |
| title_full_unstemmed | A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type |
| title_short | A Coupling Method of New EMFE and FE for Fourth-Order Partial Differential Equation of Parabolic Type |
| title_sort | coupling method of new emfe and fe for fourth order partial differential equation of parabolic type |
| url | http://dx.doi.org/10.1155/2013/787891 |
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