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Numerical Solution of Cauchy Problems for Elliptic Equations in ``Rectangle-like'' Geometries
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.ORCID-id: 0000-0002-2681-8965
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.ORCID-id: 0000-0003-2281-856X
2005 (engelsk)Inngår i: Proceedings for the FEMLAB Conference 2005, 2005Konferansepaper, Publicerat paper (Annet vitenskapelig)
Abstract [en]

We consider two dimensional inverse steady state heat conductionproblems in complex geometries. The coefficients of the elliptic equation are assumed to be non-constant. Cauchy data are given on onepart of the boundary and we want to find the solution in the wholedomain. The problem is ill--posed in the sense that the solution doesnot depend continuously on the data.

Using an orthogonal coordinate transformation the domain is mappedonto 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 theleast squares sense. This particular approximation of the derivativeis 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 amelting furnace are subject to physical and chemical wear. Temperature and heat--flux data are collected by several thermocouples locatedinside the walls. The shape of the interface between the molten ironand the walls can then be determined by solving an inverse heatconduction problem.  In our work we make extensive use of Femlab for creating testproblems. 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.

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2005.
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Identifikatorer
URN: urn:nbn:se:liu:diva-139962OAI: oai:DiVA.org:liu-139962DiVA, id: diva2:1135354
Konferanse
FEMLAB Conference, Stockholm, October 3-5, 2005.
Tilgjengelig fra: 2017-08-23 Laget: 2017-08-23 Sist oppdatert: 2017-08-30

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