Research Interests

March 2, 2005

My main research interest is the theoretical analysis and practical implementation of Finite Element Methods(FEM). FEM is a numerical method to get a finite dimensional approximation of the solution of some partial differential equations (PDEs).

PDE theory . For example, Function Spaces: Soblev space, Besov space, Holder space and more. Well-posedness of PDEs: Existence, uniqueness and continuous dependent of the data. Regularity of the solution: The smoothness of the solution will determine the approximability.

The next step is to get discrete versions of PDEs. The FEM discretization includes: to design various finite element spaces for different PDEs and to study the well-posedness of the discrete problems. The compatible discretization means discrete problems will inherit some nice properties of PDEs

Since we are approximating a function, the Approximation Theory is an essential part of FEM. The main concern is the approximability of the finite element spaces. The stability and error estimate of several projections in various norms, such as nodal interpolation, L2 projection and Galerkin projection. The properties of those projectors also depends on the regularity of the grids. For quasi-uniform grids, those questions are well studied in the literature while for nonuniform or anisotropic grids, results are much weaker.

After we get the discrete equation, we need to solve a large algebraic system in the form Ax=b. In most cases the matrix obtained by FEM is sparse and thus iterative methods is better than direct methods. Most used iterative methods in practice is CG , PCG and their variants. An most efficient iterative method is Mutligrid methods. The design and analysis of linear iterative methods can be carried in a general framework: Space decomposition and subspace correction.

Last but not least is the implementation of FEM. This procedure includes: Mesh Generation which partition the domain into grids and a robust FEM code for a class of PDEs. If possible auto generation of FEM code for different types of PDEs would be desirable.

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