# Sparse shape functions for triangular p-FEM using integrated Jacobi polynomials

## Sven Beuchler

In this talk, we investigate the following class of boundary value problems: Let be a bounded domain and let be a matrix which is symmetric and uniformly positive definite in . Find , such that

holds for all . Problem () will be discretized by means of the -version of the finite element method using triangular elements , .

Let be the reference triangle with the vertices , and and let be the isoparametric mapping to the element . On , we use integrated Jacobi polynomials as shape functions.

Then, we can show that the element stiffness matrix on has not more than nonzero entries per row if the coefficient matrix is constant on .

This result has two consequences which will be considered in this talk:

- If the coefficient matrix is piecewise constant and constant on each element and the mapping is linear, the stiffness matrix for the -FEM discretization has nonzero matrix entries, i.e. the number of nonzero matrix entries is proportionally to the number of unknowns.
- We are able to define a new preconditioner for the element stiffness matrix restricted to the interior bub bles. This preconditioner can be used as one ingredient embedded in a Domain Decomposition preconditioner of Dirichl et-Dirichlet-type.

Finally, we present some numerical examples.

The approach can be extended to the 3D case.

**REFERENCES**

[1] | S. Beuchler and J. Schöberl, `` New shape functions for triangula r -FEM using integrated Jacobi polynomials'', Report RICAM 2004-18, Johann Radon Institute for Computational and Applied Mathematics, 2004. |