Local Projection Stabilization Research Articles

Local projection stabilization virtual element method for the convection-diffusion equation with nonlinear reaction term

We present an efficient local projection-based stabilized virtual element method for the convection-diffusion equation with a nonlinear reaction term. The proposed discrete scheme contains a local projection stabilization term with a suitable virtual element stabilizer. We prove the existence and stability of the discrete solution and then the convergence estimates in the energy and L2 norms. Numerical experiments are conducted for the problems having interior and boundary layers. We validate the theoretical convergence rates and show the efficiency of the proposed method by comparing the obtained results with the residual-based stabilization methods.

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

A nodal integration based two level local projection meshfree stabilization method for convection diffusion problems

This paper presents a new two level local projection meshfree stabilization (LPMS) method under the classical stabilized conforming nodal integration (SCNI) framework to solve convection diffusion problems suitable to a linear order approximation. Dropping the higher order terms in the residual based Petrov-Galerkin stabilization methods, usual in the Gauss integration, produces erroneous results if the classical SCNI is used because of the product of a quantity and its smoothed gradient (SCNI divergence operator) in the convective term. LPMS method is an alternative way to achieve a stable solution with the linear order approximation. In the present method, the design of the stabilization of convective terms stems from the fine scales defined on the subcells of Voronoi cells of SCNI. This LPMS with SCNI combination excludes the computational need for derivatives. First order mesh basis functions are sufficient as the SCNI replaces lower order derivatives with smoothed gradients, while local projection stabilization (LPS) excludes the terms with higher order derivatives. The present LMPS method is tested against standard benchmark problems with different distorted grids and compared with three existing stabilization methods. It is found to be in good agreement with the classical numerical methods for problems with thin layers.

A local projection stabilization for convection-diffusion-reaction equations using biorthogonal systems

We consider a local projection stabilization based on biorthogonal systems for convection–diffusion–reaction differential equations with mixed boundary conditions. The approach based on biorthogonal systems is numerically more efficient than other existing approaches to obtain a uniform approximation for convection dominated problems. We prove optimal a priori error estimates for the proposed numerical technique. Numerical examples are presented to demonstrate the performance of the approach. doi: 10.1017/S1446181123000019

On a certified VMS-Smagorinsky reduced basis model with LPS pressure stabilisation

In this work we introduce a certified Reduced Basis VMS-Smagorinsky turbulence model with local projection stabilisation (LPS) on the pressure, for steady state problems. We prove its stability for Taylor-Hood discretisations of velocity-pressure or piecewise linear pressure. We construct an a posteriori error estimator for the snapshot selection with a Greedy algorithm, based on the Brezzi-Rappaz-Raviart theory of approximation of non-singular branches of non-linear PDEs. The Empirical Interpolation Method (EIM) is used for the approximation of the non-linear terms. We present some numerical tests in which we show an improved speedup on the computation of the reduced basis problem with the LPS pressure stabilisation, with respect to the method of using pressure supremizers.

Error analysis of a stable reduced order model based on the proper orthogonal decomposition method for the Allen–Cahn–Navier–Stokes system

In this paper, we develop a stable reduced order model based on the proper orthogonal decomposition method, and conduct rigorous theoretical analyses for the Allen–Cahn–Navier–Stokes system. Firstly, the full order model and the reduced order model are constructed. The full order model uses finite element discretization for space and explicit–implicit discretization for time, where the same degree of Lagrange polynomials are used to discretize phase function ϕ, velocity u and pressure p for the spatial discretization. Such spatial discretization leads to unstable schemes due to the loss of inf–sup stability condition. Therefore, we consider adding some stability terms to increase the stability of the whole system. The local projection stabilization method is added to the gradient of velocity and pressure. It should be emphasized that the local projection stabilization method relaxes the constraints of the stabilization coefficient compared with the previous research. Secondly, on the basis of full order model, we obtain the reduced order model by using proper orthogonal decomposition method. Then, we theoretically validate the energy stabilities and the error estimate of the full order model and the reduced order model, respectively. After that, some numerical experiments are performed to verify the numerical accuracy and the computational efficiency of the reduced order model.

A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers

We propose and analyze a local projection stabilization virtual element method for time-fractional Burgers equation on polygonal meshes, whose solutions display a weak singularity at the initial time. Based on the L1 scheme on a graded mesh in time and virtual element method in space, the stability and existence and uniqueness of the fully discrete scheme are proved; and the priori error estimates in L2 norm is derived with local projection stabilization term. Numerical experiments show that our method is effective for time fractional Burgers equation with high Reynolds number.

Local projection stabilisation for convection-diffusion-reaction equations using a biorthogonal system and adaptive refinement

We consider a local projection stabilisation based on biorthogonal systems and adaptive refinement for convection-diffusion-reaction differential equations. The local projection stabilisation and adaptive finite element method are both based on a biorthogonal system. We investigate the numerical efficiency of the approach when compared to the standard finite element method. Numerical examples are presented to demonstrate the performance of the approach.

A LOCAL PROJECTION STABILIZATION FOR CONVECTION–DIFFUSION–REACTION EQUATIONS USING BIORTHOGONAL SYSTEMS

AbstractWe consider a local projection stabilization based on biorthogonal systems for convection–diffusion–reaction differential equations with mixed boundary conditions. The approach based on biorthogonal systems is numerically more efficient than other existing approaches to obtain a uniform approximation for convection dominated problems. We prove optimal a priori error estimates for the proposed numerical technique. Numerical examples are presented to demonstrate the performance of the approach.

A new local projection stabilization virtual element method for the Oseen problem on polygonal meshes

A new local projection stabilization virtual element method for the Oseen problem on polygonal meshes

Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping

We study continuous finite element dicretizations for one dimensional hyperbolic partial differential equations. The main contribution of the paper is to provide a fully discrete spectral analysis, which is used to suggest optimal values of the CFL number and of the stabilization parameters involved in different types of stabilization operators. In particular, we analyze the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique, the continuous interior penalty (CIP) stabilization method and the local projection stabilization (LPS). Three different choices for the continuous finite element space are compared: Bernstein polynomials, Lagrangian polynomials on equispaced nodes, and Lagrangian polynomials on Gauss-Lobatto cubature nodes. For the last choice, we only consider inexact quadrature based on the formulas corresponding to the degrees of freedom of the element, which allows to obtain a fully diagonal mass matrix. We also compare different time stepping strategies, namely Runge-Kutta (RK), strong stability preserving RK (SSPRK) and deferred correction time integration methods. The latter allows to alleviate the computational cost as the mass matrix inversion is replaced by the high order correction iterations. To understand the effects of these choices, both time-continuous and fully discrete Fourier analysis are performed. These allow to compare all the different combinations in terms of accuracy and stability, as well as to provide suggestions for optimal values discretization parameters involved. The results are thoroughly verified numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that cubature elements combined with SSPRK and CIP or LPS stabilization are the most promising combinations.

An overlapping local projection stabilization for Galerkin approximations of Stokes and Darcy flow problems

An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming P1c finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L2-norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.

A local projection stabilization virtual element method for convection-diffusion-reaction equation

In this paper, we propose and analyze a local projection stabilization virtual element method for steady scalar convection-diffusion-reaction problem in the convective-dominated regime. Stability and optimal convergence property in the convective-dominated regime are proven in a suitable norm. Numerical results show that the proposed method is stable at very large Reynolds number and is in good agreement with the expected rate of convergence.

A local projection stabilisation finite element method for the Stokes equations using biorthogonal systems

We present a stabilised finite element method for the Stokes equations. The stabilisation is based on a biorthogonal system, which preserves the locality of the approach. We present a priori error estimates of the presented scheme and demonstrate some numerical results.

A space-time finite element method based on local projection stabilization in space and discontinuous Galerkin method in time for convection-diffusion-reaction equations

In this article, we combine the local projection stabilization (LPS) technique in space and the discontinuous Galerkin (DG) method in time to investigate the time-dependent convection-diffusion-reaction problems. This kind of stabilized space-time finite element (STFE) scheme, based on approximation space enriched by bubble functions that can increase stability, is constructed. The existence, uniqueness and stability are proved with the properties of Lagrange interpolation polynomials established on Radau points in time direction. An error estimate in L∞(L2)-norm is given by introducing the elliptic projection operators in space direction. This estimate approach is different from the previous ones that construct a special interpolant into approximation space showing an extra orthogonality property on the projection space. Since the techniques of Lagrange interpolation in time direction decouple time and space variables, the method proposed in this paper has the advantages of reducing calculation and simplifying theoretical analysis. The space and time convergence orders are illustrated in the first numerical example with smooth solutions. A comparison between the traditional STFE scheme and the constructed scheme for the problem having exponential boundary layers is presented in the second numerical example. The simulation results show that the novel method can greatly reduce nonphysical oscillations. And the influences of the stabilization parameters on the behavior of the approximate solution are discussed by some numerical results.

Error Analysis of Proper Orthogonal Decomposition Stabilized Methods for Incompressible Flows

Proper orthogonal decomposition (POD) stabilized methods for the Navier--Stokes equations are considered and analyzed. We consider two cases: the case in which the snapshots are based on a non inf-sup stable method and the case in which the snapshots are based on an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure. For the first case, we analyze a method in which the snapshots are based on a stabilized scheme with equal order polynomials for the velocity and the pressure with local projection stabilization (LPS) for the gradient of the velocity and the pressure. For the POD method we add the same kind of LPS stabilization for the gradient of the velocity and the pressure as the direct method, together with grad-div stabilization. In the second case, the snapshots are based on an inf-sup stable Galerkin method with grad-div stabilization and for the POD model we also apply grad-div stabilization. In this case, since the snapshots are discretely divergence-free, the pressure can be removed from the formulation of the POD approximation to the velocity. To approximate the pressure, needed in many engineering applications, we use a supremizer pressure recovery method. Error bounds with constants independent of inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.

Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations

Abstract Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier–Stokes equations. The spatial discretization based on inf–sup stable pairs of finite element spaces is stabilized using a one-level local projection stabilization method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for both the semidiscrete case and the fully discrete case. Numerical results support the theoretical predictions.

Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators. Furthermore, LPS is able to take advantage of the additional nodes required by degree $2p$ diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.

Local projection stabilization with discontinuous Galerkin method in time applied to convection dominated problems in time-dependent domains

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian formulation is used to handle the time-dependent domain. The stability and error estimates are given for the proposed numerical scheme. The validation of the proposed local projection stabilization scheme with higher order dG time discretization is demonstrated with appropriate numerical examples.

Simulation of viscoelastic two-phase flows with insoluble surfactants

A stabilized finite element scheme is developed for computations of buoyancy driven 3D-axisymmetric viscoelastic two-phase flows with insoluble surfactants. The numerical scheme solves the Navier–Stokes equations for the fluid flow, Giesekus constitutive equation for the effects of viscoelasticity and simultaneously an evolution equation for the surfactant concentration on the interface. The interface is tracked by the coupled arbitrary Lagrangian–Eulerian (ALE) and Lagrangian approach. The interface-resolved moving meshes allow accurate incorporation of the interfacial tension force, Marangoni forces and the jumps in the material properties. Further, the tangential gradient operator technique is used to handle the curvature approximation in a semi-implicit manner. An one-level Local Projection Stabilization (LPS), which is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh is used to stabilize the model equations. The stabilized numerical scheme allows us to use isoparametric second order conforming finite elements enriched with cubic bubble functions for velocity and viscoelastic stress, second order finite elements for surfactant concentration and discontinuous first order finite element for pressure. A number of computations are performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column with insoluble surfactants on the interface. The influence of the Marangoni number, initial surfactant concentration and Peclet number on the dynamics of the rising drop are analyzed. The numerical study shows that a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape without insoluble surfactants. The presence of insoluble surfactants forces the drop to rise slowly but the drop at the tail end is pulled up more. However, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape without insoluble surfactants. The presence of surfactants pulls the tail end of the drop up slightly and makes the tail flatter with/without small undulations depending on the magnitude of the surfactant concentrations.