scholarly journals A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries

2005 ◽  
Vol 54 (3-4) ◽  
pp. 450-469 ◽  
Author(s):  
M.J. Baines ◽  
M.E. Hubbard ◽  
P.K. Jimack
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Moving Boundaries ◽  
Time Dependent ◽  
Moving Mesh ◽  
Partial Differential ◽  
Adaptive Solution ◽  
Element Algorithm

scholarly journals A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains

2020 ◽  
Author(s):  
C M Elliott ◽  
T Ranner
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Rates Of Convergence ◽  
Time Dependent ◽  
Unified Theory ◽  
Approximation Properties ◽  
Abstract Theory ◽  
Partial Differential ◽  
Element Space

Abstract We develop a unified theory for continuous-in-time finite element discretizations of partial differential equations posed in evolving domains, including the consideration of equations posed on evolving surfaces and bulk domains, as well as coupled surface bulk systems. We use an abstract variational setting with time-dependent function spaces and abstract time-dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described, which confirm the rates of convergence.

scholarly journals Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

2011 ◽  
Vol 10 (3) ◽  
pp. 509-576 ◽  
Author(s):  
M. J. Baines ◽  
M. E. Hubbard ◽  
P. K. Jimack
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Historical Review ◽  
Porous Medium Equation ◽  
Moving Mesh ◽  
Test Problems ◽  
Partial Differential ◽  
Two Phase

AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

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