Volume 18, Issue 1

Explicit methods are generally preferred for hyperbolic problems. With this in mind the finite element method of lumped masses instead of consistent masses has been implemented to discretize the wave equation. The diagonal mass matrix resulting from lumping leads to a special system of second-order ordinary differential equations. Instead of transforming this system to an equivalent first-order system as is typically done, we solve this directly by using an explicit Runge–Kutta–Nyström method that offers improved efficiency and less memory. Explicit a posteriori error estimators developed for elliptic and parabolic problems have been extended to the wave equation. They are seen to be asymptotically exact for the method of lumped masses for second-order hyperbolic problems. Computational results indicate that the estimates are asymptotically exact.

We construct and analyze a discontinuous Galerkin–Petrov time discretization of a general evolution equation in a Hilbert space. The method is A -stable and exhibits an energy decreasing property. The approach consists in a continuous solution space and a discontinuous test space such that the time derivative of the discrete solution is contained in the test space. This is the key to get stability. We prove A -stability and optimal error estimates. Numerical results confirm the theoretical results.

A local multilevel product algorithm and its additive version are analyzed for linear systems arising from the application of adaptive finite element methods to second order elliptic boundary value problems. The abstract Schwarz theory is applied to verify uniform convergence of local multilevel methods featuring Jacobi and Gauß–Seidel smoothing only on local nodes. By this abstract theory, convergence estimates can be further derived for the hierarchical basis multigrid method and the hierarchical basis preconditioning method on locally refined meshes, where local smoothing is performed only on new nodes. Numerical experiments confirm the optimality of the suggested algorithms.