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Least-squares finite element methods for compressible Euler equations. (English) Zbl 0692.76068
Summary: A method based on backward finite differencing in time and a least- squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the $$L^ 2$$-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the $$L^ 2$$-method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted $$H^ 1$$-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

##### MSC:
 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 65N99 Numerical methods for partial differential equations, boundary value problems 76M99 Basic methods in fluid mechanics
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##### References:
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