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Geometric Analysis and Partial Differential Equations Seminar
February 14, 2013 @ 10:30 am - 11:30 pm
Dr. Bishnu Lamichane (University of Newcastle)
Some mixed finite element methods for the biharmonic problem
Finite element methods provide a powerful tool in engineering analysis and numerical solutions of boundary value problems. These methods are based on variational principle and therefore have intrinsic mathematical beauty and can be applied to complicated and nonlinear problems. The other advantage of finite element methods is that when applied to differential equations they naturally fit into the concept of so-called weak solutions in a Sobolev space. There are many situations in solving differential equations where classical solution does not exist and one has to rely on the weak solution. In the first part of the talk, we give a brief introduction to the finite element methods including the concept of weak solutions. We mainly focus on the biharmonic equation. In the second part of the talk, we consider a mixed finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. We consider two approaches: one of them is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. The second approach is based on using the gradient of the solution as an additional unknown. We prove optimal a priori estimates for both approaches.