scholarly journals MOVING-FINITE-ELEMENT SOLUTION OF TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS IN TWO SPACE DIMENSIONS

1993 ◽  
Vol 1 (2) ◽  
pp. 135-159 ◽  
Author(s):  
P. A. ZEGELING
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Dependent ◽  
Finite Element Solution ◽  
Element Solution ◽  
Partial Differential ◽  
Two Space Dimensions ◽  
Moving Finite Element

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.

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