Approximation of Cauchy problems for elliptic equations using the method of lines

In this contribution we deal with the development, theoretical examination and numerical examples of a method of lines approximation for the Cauchy problem for elliptic partial differential equations. We restrict ourselves to the Laplace equation. A more general elliptic equation containing a diffusion coefficient will be considered in a forthcoming paper. Our main results are the regularization of the illposed Cauchy problem and the proof of error estimates leading to convergence results for the method of lines. We base them principally on two major foundations. The first one is a conditional stability result for the continuous Cauchy problem. The second one consists of introducing certain 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. All proofs are carried out in detail in [1]. Key-Words: Cauchy problem, ellipitc partial differential equation, illposed problem, method of lines 1 The Cauchy problem for Poisson’s equation We consider the following Cauchy problem for Poisson’s equation on a rectangle ∆u = f in Ω = (0, 1)× (0, rmax) (1) with given boundary conditions u = fi on Σi, i = 1, 2, 3, ∂u ∂y = φ1 on Σ1, (2)