Domain decomposition approaches for the generation of equidistributing parametric curves

Boutilier, Miranda (2021) Domain decomposition approaches for the generation of equidistributing parametric curves. Masters thesis, Memorial University of Newfoundland.

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Abstract

Moving mesh methods are often used to solve boundary value problems whose solutions contain regions of rapid change. In this case, these moving mesh methods allow us to concentrate a fixed number of nodes in these regions of high variance. These meshes are obtained by solving a second order boundary value problem (BVP), which arises from the equidistribution principle. There are many real-world examples where boundary value problems are posed on curves and surfaces. Here, we focus on the case where the problem is posed on a curve that is able to be explicitly represented parametrically as x = (x₁(r), x₂(r), xₙ(r) ℝⁿ. When the solution has regions of rapid change or the curve on which the problem is posed has regions of high variance or curvature, moving mesh methods allow us to find a mesh that better resolves the function on the curve without adding additional nodes. We consider combining the solution of mesh equations with the solution of differential equations posed on parametric curves. These differential equations include both time-dependent partial differential equations (PDEs) and time-independent boundary layer problems. In addition to considering the above on a single domain, we extend these methods to form multi-domain iterations to solve these boundary value problems. Domain decomposition allows us to harness the power of parallel computing, a topic that has become popular in recent years with the increase of computing power. We provide multi-domain iterations for both time-dependent and time-independent differential equations posed on parametric curves; these include classical Schwarz and optimized Schwarz methods. These iterations are formed such that they are able to be performed in parallel. Numerical results are provided throughout to illustrate the results of the iterations. This thesis also includes theoretical results that generalize known results for classical Schwarz and optimized Schwarz methods to the case where the problem is defined on a curve.

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