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A lagrangian finite element method for the 2‐D euler equations
Author(s) -
ChaconRebollo Tomas,
Hou Thomas Y.
Publication year - 1990
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160430603
Subject(s) - mathematics , mathematical analysis , finite element method , kernel (algebra) , biot number , convergence (economics) , backward euler method , euler equations , smoothing , statistics , physics , combinatorics , mechanics , economics , thermodynamics , economic growth
A grid free Lagrangian finite element method is introduced for the 2‐D incompressible Euler equations. The method is derived based on the observation that the product of the Biot‐Savart kernel and a polynomial can be integrated analytically over any triangle. This enables us to obtain a numerically stable Lagrangian method without using numerical smoothing. Moreover, we show that the method converges uniformly with second‐order accuracy. Actually, we establish a l ∞ stability result which applies to kernels that are more singular than the Biot‐Savart kernel, as long as the kernel is L loc 1integrable. Another useful result is that we prove convergence of our method when using local regridding, which allows the method to run for longer time even with a fixed mesh. The second‐order convergence is also illustrated by our numerical experiments.