Band 9, Heft 2

We are concerned with the computation of eigenvalues of singular nonselfadjoint Sturm — Liouville problems by the method of determinants. The representation of a differential operator by an infinite matrix allows the use of Lidskii’s theorem to define its determinant. The finite section is then used to compute eigenvalues in a simple way. This direct method borrows stable methods from numerical linear algebra to compute a large number of eigenvalues with high precision. Numerical examples with nondifferentiable and complex valued coefficients are treated at the end.

In this paper, the method of lines approximation for a rather general elliptic equation containing a diffusion coefficient is considered. Our main results are the regularization of the ill-posed Cauchy problem and the proof of error estimates leading to convergence results for the method of lines. These results are based on the conditional stability of the continuous Cauchy problem and the approximation by appropriately chosen finite-dimensional spaces, onto which the possibly perturbed Cauchy data are projected. At the end of this paper, we present and discuss results of some of our numerical computations. There are multiple applications in material sciences, thermodynamics, medicine etc.; related problems are shape optimization problems which are important, e.g. for nondestructive testing, crack location, thermal tomography, and other applications.

This work presents a defect correction method with increased time accuracy. The desired time accuracy is attained with no extra computational cost. The method is applied to the evolutionary transport problem, and is proven to be unconditionally stable. In the defect step, the artificial viscosity parameter is added to the Peclet number as a stability factor, and the system is antidiffused in the correction step. The time accuracy is also increased in the correction step by modifying the right hand side.Computational results verifying the claimed space and time accuracy of the approximate solution are presented.

A survey is given of current research into the numerical solution of timeindependent systems of second-order differential equations whose diffusion coefficients are small parameters. Such problems are in general singularly perturbed. The equations in these systems may be coupled through their reaction and/or convection terms. Only numerical methods whose accuracy is guaranteed for all values of the diffusion parameters are considered here. Some new unifying results are also presented.

We obtain sharp pointwise 2nd-order estimates for both solution and derivative errors on arbitrary grids for a mimetic finite-difference approximation to solutions of one-dimensional linear boundary-value problems with separated boundary conditions. Although the scheme considered is formally inconsistent with the differential equation, it turns out to possess nice convergence properties which make it a good alternative to more standard, consistent discretizations of similar arithmetic complexity, particularly with respect to derivative errors.

A boundary value problem for a 4-th order self-adjoint ordinary differential equation is considered in the case where the coefficients of the equation and its right-hand side can be nonsmooth (discontinuous, concentrated or rapidly oscillating functions). Generalized cubic splines of deficiency 1 depending on the major coefficient of the equation are applied. An error analysis of finite element methods exploiting such splines is presented in detail including superconvergence error bounds. .