Optimal convergence of arbitrary Lagrangian–Eulerian iso-parametric finite element methods for parabolic equations in an evolving domain
Buyang Li, Yinhua Xia, Zongze Yang
Abstract
Abstract An optimal-order error estimate is presented for the arbitrary Lagrangian–Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerical solution of the Laplace equation with specified boundary value matching the velocity of the boundary. The optimal order of convergence is obtained by comparing the numerical solution with the ALE-Ritz projection of the exact solution, and by establishing an optimal-order estimate for the material derivative of the ALE-Ritz projection error.
Topics & Concepts
MathematicsMathematical analysisFinite element methodProjection (relational algebra)Laplace's equationEulerian pathBoundary value problemConvergence (economics)Ritz methodDomain (mathematical analysis)Parametric statisticsApplied mathematicsLagrangianAlgorithmPhysicsEconomicsStatisticsThermodynamicsEconomic growthAdvanced Numerical Methods in Computational MathematicsAdvanced Mathematical Modeling in EngineeringNumerical methods in engineering