A coupling method based on new MFE and FE for fourth-order parabolic equation
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In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L 2 and H 1-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L 2)2-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.
KeywordsFourth-order parabolic equation New coupling method New mixed scheme Finite element scheme Square integrable (L2(Ω))2 space Error estimates
Mathematics Subject Classification65M60 65N15 65N30
The authors thank the anonymous referees and editors for their helpful comments and suggestions, which greatly improve the article. This work is supported by National Natural Science Fund (11061021), Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106, 2011BS0102), Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJ10006, NJ10016), Program of Higher-level talents of Inner Mongolia University (125119, Z200901004, 30105-125132).