Numerical Solution of Cauchy Problems for Elliptic Equations in "Rectangle-like" Geometries
2005 (English)In: FEMLAB Conference,2005, Stockholm: Comsol AB , 2005Conference paper (Other academic)
We consider two dimensional inverse steady state heat conduction
problems in complex geometries. The coefficients of the elliptic
equation are assumed to be non-constant. Cauchy data are given on one
part of the boundary and we want to find the solution in the whole
domain. The problem is ill--posed in the sense that the solution does
not depend continuously on the data.
Using an orthogonal coordinate transformation the domain is mapped
onto a rectangle. The Cauchy problem can then be solved by replacing
one derivative by a bounded approximation. The resulting well--posed
problem can then be solved by a method of lines.
A bounded approximation of the derivative can be obtained by
differentiating a cubic spline, that approximate the function in the
least squares sense. This particular approximation of the derivative
is computationally efficient and flexible in the sense that its easy
to handle different kinds of boundary conditions.
This inverse problem arises in iron production, where the walls of a
melting furnace are subject to physical and chemical wear. Temperature
and heat--flux data are collected by several thermocouples located
inside the walls. The shape of the interface between the molten iron
and the walls can then be determined by solving an inverse heat
In our work we make extensive use of Femlab for creating test
problems. By using Femlab we solve relatively complex model problems
for the purpose of creating numerical test data used for validating
our methods. For the types of problems we are intressted in numerical
artefacts appear, near corners in the domain, in the gradients that
Femlab calculates. We demonstrate why this happen and also how we
deal with the problem.
Place, publisher, year, edition, pages
Stockholm: Comsol AB , 2005.
Ill-posed, Cauchy Problem, Elliptic Equation
IdentifiersURN: urn:nbn:se:liu:diva-29471Local ID: 14820OAI: oai:DiVA.org:liu-29471DiVA: diva2:250286