First-Order System LL* (FOSLL*) for General Scalar Elliptic Problems in the Plane

Abstract

This paper develops new first-order system LL* (FOSLL*) formulations for scalar elliptic partial differential equations. It extends the work of [Z. Cal et al., SIAM J. Numer. Anal., 39 (2001), pp. 1418--1445], where the FOSLL* methodology was first introduced. One focus of that paper was to develop \FL formulations that allow the use of H1-conforming finite element spaces and optimal multigrid solution techniques to construct L2 approximations of the dependent variables in the presence of discontinuous coefficients. The problems for which this goal was achieved were limited to those with no reaction term and with Dirichlet and Neumann boundaries that were individually connected; that is, each had at most one component. Here, new FOSLL* formulations are developed to achieve the same goals on a wider class of problems, including problems with reaction terms, Dirichlet and Neumann boundaries with multiple components, reentrant corners, and points at which Dirichlet and Neumann boundaries meet with an inner angle greater than $\pi/2$. The efficiency of the improved FOSLL* formulations is illustrated by a series of numerical examples.