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A lagrangian finite element method for the 2‐D euler equations

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 integrable. 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.