A Multi-Level Iterative Method For Solving Finite Element Equations

Abstract

Abstract Subroutine PLTMG is a Fortran program for solving self-adjoint elliptic boundary value problems in general regions of R2. It is based on a piecewise linear-triangle finite element method, an adaptive grid refinement procedure, and a multi-level method to solve the resulting sets of linear equations. In this work we describe the method, and present some numerical results and comparisons. Introduction Consider the model elliptic boundary value problem problem(1.1) where the coefficient a(x,y) (b(x,y)) is positive (nonnegative) in omega. In this work we discuss the performance of a program which solves (1.1) using a performance of a program which solves (1.1) using a Rayleigh-Ritz-Galerkin method based on piecewise-linear triangular finite elements, a multi-level piecewise-linear triangular finite elements, a multi-level iterative scheme for solving the resulting matrix equations, and an adaptive grid refinement procedure. A more detailed discussion of the program appears in [14]. For expositional convenience, we assume g1 + g2 = 0, and that omega is a polygon, although our FORTRAN subroutine PLTMG is designed for the more general equation (1.1). In the Rayleigh-Ritz-Galerkin procedure, we seek an approximate solution to the weak form of (1.1) (1.2) where and (,) denotes the usual L2(omega) inner product. We use (omega) (omega) to denote the subspace of the usual Sobolev space (omega) whose elements satisfy essential boundary conditions. Associated with the bilinear form a(,) is the energy norm= a(u,u). Let denote a triangulation of omega, and let(omega) denote the N-dimensional space of C piecewise-linear polynomials associated with. The piecewise-linear polynomials associated with. The finite element approximation of u in (1.2) is the function epsilon which satisfies (1.3) Once a basis for has been selected (the nodal basis is usually chosen), (1.3) can be reformulated as a system of linear equations (1.4) where Aij = a(phi j, phi i), and Fi = (f, phi i). Usually N is large and the stiffness matrix A is sparse. Our multi-level solution procedure involves a sequence of nested triangulations, with nested in, and the corresponding Nj-dimensional subspaces,, of C0 piecewise-linear polynomials. (By nested, we mean that each triangle polynomials. (By nested, we mean that each triangle of intersects the interior of exactly one triangle of. Corresponding to each subspace N is the problem Pj, the analog of (1.3): Find uj epsilon Mj satisfying (1.5) a(uj,v)=(f,v) for all v epsilon j.