The most familiar family of finite elements is the Lagrange family, which provide the canonical finite element approximation of H1 on simplicial meshes in any dimension. In this talk we discuss families of simplicial and cubical finite elements—some previously known and some new—which are natural extensions of the Lagrange family in various ways. Even for some of the long known elements, a modern viewpoint based on the finite element exterior calculus provides new properties and insights.

We consider a Cartesian PML approximation to solutions of acoustic scattering problems on an unbounded domain in ℝ² and ℝ³. The perfectly matched layer (PML) technique modifies the equations outside of a bounded domain containing the region of interest. This is done in such a way that the new problem (still on an unbounded domain) has a solution which agrees with the solution of the original problem. The new problem has a solution which decays much faster, thus suggesting replacing it by a problem on a bounded domain. The perfectly matched layer (PML) technique, in a curvilinear coordinate system and in Cartesian coordinates, has been studied for acoustic scattering applications both in theory and computation. Using a different approach we extend the results of Kim and Pasciak concerning the PML technique in Cartesian coordinates. The exponential convergence of approximate solutions as a function of domain size and/or the PML "strength" parameter, σ₀ is also shown. We note that once the stability and convergence of the (continuous) truncated problem has been established, the analysis of the resulting finite element approximations is then classical. Finally, the results of numerical computations illustrating the theory, in terms of efficiency and parameter dependence of the Cartesian PML approach will be given.

We study the lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. We first recall that the maximum principle for the heat equation does not carry over to the the spatially semidiscrete standard Galerkin finite element method, using continuous, piecewise linear approximating functions. However, for the lumped mass variant the situation is more advantageous. We present necessary and sufficient conditions on the triangulation, expressed in terms of properties of the stiffness matrix, for the semidiscrete lumped mass solution operator to be a positive operator or a contraction in the maximum-norm.

We then turn to error estimates in the L₂-norm. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method, whereas nonsmooth initial data estimates require special assumptions on the triangulations.

We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods.