Design and analysis of numerical methods for free- and moving-boundary problems
- This thesis presents the design and analysis of numerical methods for free- and moving-boundary problems: partial differential equations posed on domains that change with time. Two principal developments are presented. First, a novel framework is introduced for solving free- and moving-boundary problems with a high order of accuracy. This framework has the distinct advantage that it can handle large domain deformations easily (a common difficulty faced by conventional deforming-mesh methods) while representing the geometry of the moving domain exactly (an infeasible task for conventional fixed-mesh methods). This is accomplished using a universal mesh: a background mesh that contains the moving domain and conforms to its geometry at all times by perturbing a small number of nodes in a neighborhood of the moving boundary. The resulting framework admits, in a general fashion, the construction of methods that are of arbitrarily high order of accuracy in space and time when the boundary evolution is prescribed. Numerical examples involving phase-change problems, fluid flow around moving obstacles, and free-surface flows are presented to illustrate the technique. Second, a unified analytical framework is developed for establishing the convergence properties of a wide class of numerical methods for moving-boundary problems. This class includes, as special cases, the technique described above as well as conventional deforming-mesh methods (commonly known as arbitrary Lagrangian-Eulerian, or ALE, schemes). An instrumental tool developed in this analysis is an abstract estimate, which applies to rather general mesh motions, for the error incurred by finite element discretizations of parabolic moving-boundary problems. Specializing the abstract estimate to particular choices of the mesh motion strategy and finite element space leads to error estimates in terms of the mesh spacing for various semidiscrete schemes. We illustrate this by deriving error estimates for ALE schemes under mild assumptions on the nature of the mesh deformation and the regularity of the exact solution and the moving domain, and we do the same for universal meshes.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Gawlik, Evan Scott
|Stanford University, Institute for Computational and Mathematical Engineering.
|Statement of responsibility
|Evan Scott Gawlik.
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2015.
- © 2015 by Evan Scott Gawlik
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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