Finite difference methods for singularly perturbed problems on non-rectangular domains
Dunne, Raymond K. (2005) Finite difference methods for singularly perturbed problems on non-rectangular domains. PhD thesis, Dublin City University.
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Singularly perturbed problems arise in many branches of science and are characterised mathematically by the presence of a small parameter m u ltip ly in g one or more of the highest derivatives in a differential equation. This thesis concerns singularly perturbed problems posed on non-rectangular domains. The methodology used is to perform a co-ordinate transformation to pose the problem on a rectangular domain and to then study the transformed problem.
We first consider a class of parabolic problems. We classify the problems in the transformed problem class according to the nature and location of the layers present in th e ir solution. This classification then enables us to design numerical methods specific to each class of problems. Known theoretical results are stated for the convergence of some of the methods. We then examine in detail one particular method. Under certain assumptions it is shown that the numerical solutions generated by the method converge uniformly with respect to the singularly perturb ed parameter. Detailed numerical results are then presented which verify the theoretical results.
The next class of problems considered is a class of elliptic problems. In this case the transformed differential equation contains a new term and the situation is thus more complex. For this reason we consider only the case when regular layers are present. An appropriate numerical method is constructed and under various assumptions it is proved th a t the numerical solutions converge uniformly, in the perturbed case, i.e., when the singularly perturbed parameter is small. This is the central result of the thesis. Extensive numerical computations are presented which verify the theoretical result.
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