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 $ \Omega\subset\mathbbm{R}^{2}$ be a bounded domain and let $ \mathcal{A}(x,y)$ be a matrix which is symmetric and uniformly positive definite in $ \Omega$ . Find $ u\in H^1_{\Gamma_1}(\Omega)=\{u\in H^1(\Omega), u=0 \;\textrm{on}\; \Gamma_1 \}$ $ \Gamma_1\cap\Gamma_2=\emptyset$ , $ \Gamma_1\cup\Gamma_2= \partial\Omega$ such that

$\displaystyle a_\triangle(u,v):=\int_{\Omega} (\nabla u)^T \mathcal{A}(x,y) \na... ...\Gamma_2} f_1 v := \langle f,v \rangle_\Omega +\langle f_1,v \rangle_{\Gamma_2}$ (1)

holds for all $ v\in H^1_{\Gamma_1}(\Omega)$ . Problem ([*]) will be discretized by means of the $ hp$ -version of the finite element method using triangular elements $ \triangle_s$ , $ s=1,\ldots,nel$ .

Let $ \hat{\triangle}$ be the reference triangle with the vertices $ (-1,-1)$ , $ (1,-1)$ and $ (0,1)$ and let $ F_{s}: \hat{\triangle}\to \triangle_{s}$ be the isoparametric mapping to the element $ \triangle_{s}$ . On $ \hat{\triangle}$ , we use integrated Jacobi polynomials as shape functions.

Then, we can show that the element stiffness matrix on $ \hat{\triangle}$ has not more than $ 25$ nonzero entries per row if the coefficient matrix is constant on $ \hat{\triangle}$ .

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

  1. If the coefficient matrix $ \mathcal{A}(x,y)$ is piecewise constant and constant on each element and the mapping $ F_{s}$ is linear, the stiffness matrix for the $ hp$ -FEM discretization has $ \mathcal{O}(p^2)$ nonzero matrix entries, i.e. the number of nonzero matrix entries is proportionally to the number of unknowns.
  2. 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 $ p$ -FEM using integrated Jacobi polynomials'', Report RICAM 2004-18, Johann Radon Institute for Computational and Applied Mathematics, 2004.