FINITE ELEMENT LEAST SQUARES SOLUTION OF THE EULER EQUATIONS USING LINEAR AND QUADRATIC APPROXIMATIONS

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Abstract
We present a method based on time differencing and a least squares finite element approximation for the solution of the compressible Euler equations. The scheme is implicit and unconditionally stable and has been implemented using both linear and quadratic triangular elements in two dimensions. The method is based upon the minimization of the L2 norm of the equation residuals at each time step. The resulting system of equations is symmetric positive-definite and is solved using an incomplete Choleski conjugate gradient algorithm. The least squares residuals provide an estimation of the the error and this is used to adaptively refine the mesh and hence to improve the quality of the computed solution. Several numerical examples are included which demonstrate the numerical performance of the approach when it is used to solve problems of compressible inviscid flows on unstructured triangular meshes.