Least-Squares Finite Element Methods and Algebraic Multigrid Solvers for Linear Hyperbolic PDEs

Abstract

Least-squares finite element methods (LSFEMs) for scalar linear partial differential equations (PDEs) of hyperbolic type are studied. The space of admissible boundary data is identified precisely, and a trace theorem and a Poincaré inequality are formulated. The PDE is restated as the minimization of a least-squares functional, and the well-posedness of the associated weak formulation is proved. Finite element convergence is proved for conforming and nonconforming (discontinuous) LSFEMs that are similar to previously proposed methods but for which no rigorous convergence proofs have been given in the literature. Convergence properties and solution quality for discontinuous solutions are investigated in detail for finite elements of increasing polynomial degree on triangular and quadrilateral meshes and for the general case that the discontinuity is not aligned with the computational mesh. Our numerical studies found that higher-order elements yield slightly better convergence properties when measured in terms of the number of degrees of freedom. Standard algebraic multigrid methods that are known to be optimal for large classes of elliptic PDEs are applied without modifications to the linear systems that result from the hyperbolic LSFEM formulations. They are found to yield complexity that grows only slowly relative to the size of the linear systems.