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.

Abstract

We present the design of a high-resolution Petrov–Galerkin (HRPG) method using linear finite elements
for the problem defined by the residual

R (phi):= partial phi /partial t + u partial phi /partial x - k partial^2 phi /partial x^2 +s phi -f

where k; s > o = 0. The structure of the method in 1D is identical to the consistent approximate upwind Petrov–
Galerkin (CAU/PG) method [A.C. Galeão, E.G. Dutra do Carmo, A consistent approximate upwind Petrov–
Galerkin method for the convection-dominated problems, Comput. Methods Appl. Mech. Engrg. 68
(1988) 83–95] except for the definitions of the stabilization parameters. Such a structure may also be attained
via the finite-calculus (FIC) procedure [E. Oñate, Derivation of stabilized equations for numerical
solution of advective–diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Engrg.
151 (1998) 233–265; E. Oñate, J. Miquel, G. Hauke, Stabilized formulation for the advection–diffusion–
absorption equation using finite-calculus and linear finite elements, Comput. Methods Appl. Mech. Engrg.
195 (2006) 3926–3946] 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 nonregular regimes. The design procedure 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. It is shown that the method indeed reproduces stabilized
high-resolution numerical solutions for a wide range of values of u; k; s and f . Finally, some remarks
are made on the extension of the HRPG method to multidimensions.

Abstract

A multidimensional extension of the HRPG method using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction–diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors li and the stabilization parameters αi, βi are calculated along these li. Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of αi, βi are a direct extension of their counterparts in 1D. Using αi, βiand li, objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective—stabilization with high-resolution

Abstract

The paper addresses the development of time-accurate methods for solving transient convection-diffusion-reaction problems using finite elements. Multi-stage time-stepping schemes of high accuracy are used. They are first combined with a Galerkin formulation to briefly recall the time-space discretization. Then spatial stabilization techniques are combined with high-order time-stepping schemes. Moreover, a least-squares formulation is also developed for these high-order time schemes combined with C0 finite elements (in spite of the diffusion operator and without reducing the strong form into a system of first-order differential equations). The weak forms induced by the SUPG, GLS, SGS and least-squares formulations are presented and compared. In a companion paper (Part II of this work), the phase and damping properties of the developed schemes are analysed and numerical examples are included to confirm the effectiveness of the proposed methodology for solving time-dependent convection-diffusion-reaction problems.

Abstract

The paper addresses the development of time‐accurate methods for solving transient convection–diffusion –reaction problems using finite elements. The accuracy characteristics of the spatially stabilized implicit multi‐stage time‐stepping schemes developed in a companion paper (Part I of this work) are analysed and compared here. This is done by means of a Fourier analysis. An important improvement is observed when the order of the method is increased. Moreover, the stabilization techniques proposed (streamline‐upwind Petrov–Galerkin (SUPG), Galerkin least‐square (GLS), sub‐grid scale (SGS) and least squares) do not degrade the phase accuracy. Finally, some examples are presented to show the applicability of these schemes. 

Abstract

We present a procedure for coupling the uid and transport equations to model the distribution of a pollutant in a street canyon, in this case, black carbon (BC). The fluid flow is calculated with a stabilized finite element method using the Quasi-Static Variational Multiscale (QS-VMS) technique. For the temperature and pollutant transport we use a semi-Lagrangian procedure, based on the Particle Finite Element Method (PFEM) combined with an Eulerian method based on a Finite Increment Calculus (FIC) formulation. Both methods are implemented on the open-source KRATOS Multiphysics platform. The coupled numerical formulation is applied to the prediction of the transport of BC in a street canyon, which can be a useful tool to lessen the impact of pollutants on pedestrians. Two test cases have been studied: a 2D simplified case and a more complex 3D one. The main goal of this study is to propose a useful tool to study the effect of pollution on pedestrians in a street-level scale. Good comparison with experimental results is obtained.

Abstract

In this paper we present an accurate stabilized FIC-FEM formulation for the 1D advection-diffusion-reaction equation in the exponential and propagation regimes using two stabilization parameters. Both the steady-state and transient solutions are considered. The stabilized formulation is based on the standard Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method. The steady-state problem is considered first. The optimal value of the two stabilization parameters ensuring an exact (nodal) FEM solution using uniform meshes of linear 2-noded elements is obtained. In the absence of the absorption term the formulation simplifies to the standard one-parameter Petrov-Galerkin method for the advection-diffusion problem. For the diffusion-reaction case one stabilization parameter is just needed and the diffusion-type stabilization term is identical to that obtained by Felippa and O˜ nate [16] using a variational FIC approach. A procedure for computing the stabilization parameters for the transient problem is proposed. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of steady-state and transient 1D advection-diffusion-radiation problems for a the range of physical parameters and boundary conditions. Finally we outline the procedure to extend the 1D FIC-FEM formulation to multidimensions.