Least‐squares finite element methods for compressible Euler equations

Abstract

SUMMARYA method based on backward finite differencing in time and a least-squares finite element scheme for firstorder systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L2-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 L2-method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted If'-norm of the residual is proposed and leads to a successful scheme with highdegree 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. KEY WORDS Compressible Euler equations Finite element Least-squares method Shock resolution

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

“…Consequently, most finite element models of the Navier-Stokes equations based on the weak-form Galerkin procedure do not guarantee the minimization of the error in the solution or in the differential equation. Least-squares finite element models offers an appealing alternative to the commonly used weak-form Galerkin procedure for fluids and have received substantial attention in the academic literature in recent years (see, for example, [21,22,24,31,33,28,27,36,37,35,30]). The least-squares formulation allows for the construction of finite element models for fluids that, when combined with high-order finite element technology [22,4,5,38,17,29,31,31,49] possess many of the attractive qualities associated with the well-known Ritz method [43] such as global minimization, best approximation with respect to a well-defined norm, and symmetric positive-definiteness of the resulting finite element coefficient matrix [9].…”

Least-squares finite element method for fluid dynamics

“…However, the implementation is not so easy. Thus, we proposed an alternative technique in which the objective function is modified to control the residual derivatives also [63][64][65]. Accordingly, the following multi-objective optimization problem is constructed [66] :…”

“…For the numerical solution of the one-and two-dimensional compressible Euler equations, we proposed a class of least-squares methods [63,64]. We begin by considering the first-order implicit time-differenced non-conservative formulation.…”

Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

“…In the least-squares approach the test function W in (24) is replaced by the variation of the residual 271 of the governing equations [10,11]. Conceptually this is equivalent to minimizing the residual in a least-squares sense.…”