Balancing domain decomposition for mixed finite elements

Abstract

The rate of convergence of the Balancing Domain Decomposition method applied to the mixed finite element discretization of second-order elliptic equations is analyzed. The Balancing Domain Decomposition method, introduced recently by Mandel, is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet data, a local problem with Neumann data, and a "coarse grid" problem to propagate information globally and to insure the consistency of the Neumann problems. It is shown that the condition number grows at worst like the logarithm squared of the ratio of the subdomain size to the element size, in both two and three dimensions and for elements of arbitrary order. The bounds are uniform with respect to coefficient jumps of arbitrary size between subdomains. The key component of our analysis is the demonstration of an equivalence between the norm induced by the bilinear form on the interface and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 slash 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{1/2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm of an interpolant of the boundary data. Computational results from a message-passing parallel implementation on an INTEL-Delta machine demonstrate the scalability properties of the method and show almost optimal linear observed speed-up for up to 64 processors.