Volume 17, Issue 2

We develop new a posteriori error estimates for the P 1 finite element approximation of the diffusion equation with an arbitrary piecewise constant tensor K . The estimates are established for a special composite norm of the error that is formed by the energy norm of the solution error and the K –1/2 -weighted L 2 -norm of the flux error.

A regularization method is proposed for the polynomial approximation of a function from its approximated values in a fixed family of nodes. As a regularization parameter we consider the number of nodes. We present explicit expressions for the optimal number of nodes in terms of the original error of the approximated values of the function. These problems appear frequently in studying inverse problems and when a smoothing technique should be applied to a series of numerical data. We obtain estimation of the approximation error by means of discrete versions of a convolution operators with polynomial kernels, and we observe the differences between the use of positive and non positive kernels. Some numerical examples are provided to illustrate the efficiency and computational performance of the method. They also help us to compare different criteria for the construction of polynomial approximations.

We propose a nonconforming spectral/ hp element method for solving elliptic systems on non smooth domains using parallel computers. A geometric mesh is used in a neighbourhood of the corners and a modified set of polar coordinates, as defined by Kondratiev [Diff. Equations 6: 1392 – 1401, 1970], is introduced in these neighbourhoods. In the remaining part of the domain Cartesian coordinates are used. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. The set of common boundary values consists only of the values of the spectral element functions at the vertices of the polygonal domain. Since the cardinality of the set of common boundary values is so small, a nearly exact Schur complement matrix can be computed. The method is exponentially accurate and asymptotically faster than the h - p finite element method. The normal equations obtained from the least-squares formulation can be solved by the preconditioned conjugate gradient method using a parallel preconditioner. The algorithm is implemented on a distributed memory parallel computer with small inter- processor communication. Numerical results for scalar problems and the equations of elasticity are provided to validate the error estimates and estimates of computational complexity that have been obtained.

This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems.