On Some Applications of the H {stable Wavelet{like Hierarchical Finite Element Space Decompositions

In this paper we rst review the construction of stable Riesz bases for nite element spaces with respect to Sobolev norms. Then, we construct optimal order multilevel preconditioners for the matrices in the normal form of the equations arising in the nite element discretization of non{symmetric second order elliptic equations. The optimality of the AMLI methods is proven under H 2 {regularity assumption on underlined elliptic problem. A second application is in the area of domain embedding methods that utilizes H 1 {bounded extension operators of data, nite element functions deened on the boundary across which the given (irregular) domain is embedded into a more regular (e.g., parallelepiped) one. 1. Introduction In this paper we are concerned with the construction of eecient numerical methods for matrix problems arising from nite element methods for elliptic partial diierential equations. In practical computation, the standard nodal basis for the nite element space is often chosen as the computational basis and the resulting matrices are ill-conditioned. In Vassilevski and Wang VW97] the objective was to seek a substitution for the standard nodal basis so that the stiiness matrix arising from the new basis is well{conditioned, preserving the two major properties required for a computationally feasible basis: (a) the basis functions must be computable and (b) they must also have local support, hence the resulting stiiness matrix is sparse. This paper will rst review the construction of local projections operators by the wavelet-like method proposed in VW96a]. The latter wavelet{like projections operators have a main application in the construction of a stable Riesz basis with the above mentioned features for the nite element application to elliptic problems. Attempts in the search of a stable Riesz basis with some restrictions, either on the mesh or on the analysis, have been made in Griebel and Oswald GO94], Kotyczka and Oswald KO95], and Stevenson Ste95a], Ste95b]. For a comparative study on the construction of economical Riesz bases for Sobolev spaces we refer to Lorentz and Oswald LO96]. The method from Vassilevski and Wang VW96a], VW97] is general and provides a satisfactory answer for most of the elliptic equations. It is based on modifying the existing (unstable) hierarchical basis by using operators which are approximations of the L 2 {projections onto coarse nite element spaces. For more