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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This entry was posted on March 2, 2005 at 11:07 pm and is filed under Research.

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