# An accurate and efficient finite element Euler equation algorithm

## Abstract

An implicit finite element algorithm is established for numerical solution of the multi-dimensional Euler equations. The theoretical construction employs a multi-pole expansion of the Galerkin-Weighted Residuals concept requiring the error in the semi-discrete (finite element) approximation to be rendered orthogonal to the finite dimensional sub space of H^{1} selected to generate the solution. Identification of the tensor matrix product resolution of the Newton iteration algorithm Jacobian, valid for arbitrary degree k of the approximation subspace, and a generalized-coordinates framework amenable to use with any regularizing coordinate transformation, renders the algorithm economically competitive. Numerical results document accurate solutions for mixed subsonic-supersonic flows, including robust resolution of shocks on rather coarse discretizations of R^{n}, as well as convergence estimates in H^{1} and E. The algorithm concept is equally applicable to the complete Navier-Stokes equations.

## Keywords

Computational Fluid Dynamic Jacobian Formulation Finite Element Algorithm Coarse Discretizations Computational Fluid Dynamic Research## Preview

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