HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980205-A

Introduction

The preconditioned conjugate gradient (PCG) based domain decomposition methods are intensively investigated parallel algorithms for the numerical solution of elliptic problems. Numerous papers are devoted to this topic, see for example [1,2,3,4,5,7].

Several Schur complement preconditioner constructions are an appropriate matrix representation of the $H^{1/2}$ seminorm [1,7]. In this paper we give a new explicit circulant representation of the $H^{1/2}$ seminorm in the space of bilinear finite elements defined on the surface of the unit cube

$$U = [0,1]^3.$$ (1)

Our construction is based on the separability property of the $H^{1/2}$ seminorm on rectangules and an application of a sparse matrix representation [6] of the one-dimensional $H^{1/2}$ seminorm in the space of linear finite elements. The given construction can be generalized in a straightforward way to more complicated cases, for example complex domains that are unions of rectangular elements.

This paper is organised as follows. The separability property of the $H^{1/2}$ seminorm on a rectangular shaped surface is discussed in Section 2. Section 3 contains the description of a representation of the 'partial' seminorms by a sum of one-dimensional seminorms in the space of bilinear finite elements. The circulant matrix representation of the $H^{1/2}$ seminorm on the surface of $U$ (1) is described in Section 4. Numerical results regarding to this representation are given in Section 5.

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980205-A