Abstract

We present three new stabilized finite element (FE) based Petrov–Galerkin methods for the convection–diffusion–reaction (CDR), the Helmholtz and the Stokes problems, respectively. The work embarks upon a priori analysis of some consistency recovery procedures for some stabilization methods belonging to the Petrov–Galerkin framework.

It was found that the use of some standard practices (e. g.M-Matrices theory) for the design of essentially non-oscillatory numerical methods is not appropriate when consistency recovery methods are employed. Hence, with respect to convective stabilization, such recovery methods are not preferred. Next, we present the design of a high-resolution Petrov–Galerkin (HRPG) method for the CDR problem. The structure
of the method in 1d is identical to the consistent approximate upwind (CAU) Petrov–Galerkin method [68] except for the definitions of the stabilization parameters.

Such a structure may also be attained via the Finite Calculus (FIC) procedure [141] by an appropriate definition of the characteristic length. The prefix high-resolution is used here in the sense popularized by Harten, i. e.second order accuracy for smooth/regular regimes and good shock-capturing in non-regular regimes. The design procedure in 1d embarks on the problem of circumventing the Gibbs phenomenon observed in L2 projections. Next, we study the conditions on the stabilization parameters to circumvent the global oscillations due to the convective term. A conjuncture of the two results is made to deal with the problem at hand that is usually plagued by Gibbs, global and dispersive oscillations in the numerical solution. A multi dimensional extension of the HRPG method using multi-linear block finite elements is also presented.
Next, we propose a higher-order compact scheme (involving two parameters) on structured meshes for the Helmholtz equation. Making the parameters equal, we recover the alpha-interpolation of the Galerkin finite element method (FEM) and the classical central finite difference method. In 1d this scheme is identical to the alphainterpolation method [140] and in 2d choosing the value 0.5 for both the parameters, we recover the generalized fourth-order compact Padé approximation [81, 168] (therein using the parameter
gamma = 2). We follow [10] for the analysis of this scheme and its performance on square meshes is compared with that of the quasi-stabilized FEM [10]. Generic expressions for the parameters are given that guarantees a dispersion accuracy of sixth-order should the parameters be distinct and fourth-order should they be equal. In the later case, an expression for the parameter is given that minimizes the maximum relative phase error in 2d. A Petrov–Galerkin formulation that yields the aforesaid scheme on structured meshes is also presented. Convergence studies of the
error in the L2 norm, the H1 semi-norm and the l1 Euclidean norm is done and the pollution effect is found to be small.

Finally, we present a collection of stabilized FE methods derived via first-order and second-order FIC procedures for the Stokes problem. It is shown that several well known existing stabilized FE methods such as the penalty technique, the Galerkin Least Square (GLS) method [93], the Pressure Gradient Projection (PGP) method [35] and the orthogonal sub-scales (OSS) method [34] are recovered from the general
iii residual-based FIC stabilized form. A new family of Pressure Laplacian Stabilization (PLS) FE methods with consistent nonlinear forms of the stabilization parameters are derived. The distinct feature of the family of PLS methods is that they are nonlinear and residual-based, i. e.the stabilization terms depend on the discrete residuals of the momentum and/or the incompressibility equations. The advantages and disadvantages of these stabilization techniques are discussed and several examples of application are presented.