Volume 19, Issue 1

We study in this paper a P1 finite element approximation of the solution in of a biharmonic problem. Since the P1 finite element method only leads to an approximate solution in , a discrete Laplace operator is used in the numerical scheme. The convergence of the method is shown, for the general case of a solution with regularity, thanks to compactness results and to the use of a particular interpolation of regular functions with compact supports. An error estimate is proved in the case where the solution is in . The order of this error estimate is equal to 1 if the solution has a compact support, and only 1/5 otherwise. Numerical results show that these orders are not sharp in particular situations.

Recently, Hofreither, Langer and Pechstein have analyzed a nonstandard finite element method based on element-local boundary integral operators. The method is able to treat general polyhedral meshes and employs locally PDE-harmonic trial functions. In the previous work, the primal formulation of the method has been analyzed as an inexact Galerkin scheme, obtaining H 1 error estimates. In this work, we pass to an equivalent mixed formulation. This allows us to derive error estimates in the L 2 -norm, which were so far not available. Many technical tools from our previous analysis remain applicable in this setting.

We discuss numerical properties of continuous Galerkin–Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.

We give a theorem on error estimates of approximate solutions for the ordinary functional differential equation. The error is estimated by a solution of an initial problem for nonlinear differential functional equation. We apply this general result to the investigation of the convergence of the numerical method of lines generated by evolution functional differential equations. Initial boundary value problems for Hamilton Jacobi functional differential equations and parabolic functional differential problems are considered. Nonlinear estimates of the Perron type with respect to the functional variable for given operators are assumed.