Moving mesh methods for problems with layer phenomena
Hill, Róisín
Hill, Róisín
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Publication Date
2022-06-10
Type
Thesis
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Abstract
This thesis is concerned with developing, implementing, testing and refining novel algorithms for generating layer-adapted meshes for the numerical solution of singularly perturbed differential equations (SPDEs). We consider a variety of problems: ordinary and partial differential equations, time-dependent and stationary, scalar and coupled, reactiondiffusion and convection-diffusion. We also construct meshes of different types: a priori fitted and a posteriori fitted. However, there is a common thread running through the thesis: the algorithms we propose are all based around the concept of mesh partial differential equations (MPDEs). We concentrate on differential equations whose solutions contain boundary or interior layers; these are particularly sensitive and require concentrated meshes to accurately capture these features. Moving meshes partial differential equations [Huang and Russell, 2011] generate meshes that evolve in time for time-dependent problems. We extend this idea to generate meshes using MPDEs for both one- and two-dimensional stationery singularly perturbed differential equations whose solutions exhibit boundary layers. The MPDE is nonlinear, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh intervals. We resolve this by proposing a new nonlinear solver based on h-refinement. The main advantage of the MPDE method for generating layer-adapted grids is that, unlike conventional schemes, they are not restricted to tensor-product grids. Therefore, it is feasible to solve problems whose solutions have layers that vary in width spatially or that are posed on irregularly shaped domains. Specifically, this new approach generalises the concept of a Bakhvalov mesh. In that setting, a priori asymptotic information on the SPDE’s solution, and its derivatives, is used. We further extend the framework to using a posteriori estimates for those derivatives, leading to more truly adaptive meshes. All algorithms presented have been coded in FEniCS [Alnæs et al., 2015], an opensource software system for generating finite element solutions. Key aspects of the code are included, and published on https://osf.io/dpexh/, to demonstrate the reproducibility of our results, and to make them accessible to other researchers.
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NUI Galway