A posteriori error estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H¹ norm and pressure in L² norm are derived. Furthermore, a global upper bound of u ? u_h in L²‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.     

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

This work combines two complementary strategies for solving the steady incompressible Navier–Stokes model with a zeroth‐order term, namely, a stabilized finite element method and a mesh–refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth‐order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

An adaptive boundary element method for the exterior Stokes problem in three dimensions

^** Email: vjervin{at}clemson.edu^*** Email: norbert.heuer{at}brunel.ac.uk We present an adaptive refinement strategy for the h-versionof the boundary element method with weakly singular operatorson surfaces. The model problem deals with the exterior Stokesproblem, and thus considers vector functions. Our error indicatorsare computed by local projections onto 1D subspaces definedby mesh refinement. These indicators measure the error separatelyfor the vector components and allow for component-independentadaption. Assuming a saturation condition, the indicators giverise to an efficient and reliable error estimator. Also we describehow to deal with meshes containing quadrilaterals which arenot shape regular. The theoretical results are underlined bynumerical experiments. To justify the saturation assumption,in the Appendix we prove optimal lower a priori error estimatesfor edge singularities on uniform and graded meshes.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

A quadratic equal‐order stabilized method for Stokes problem based on two local Gauss integrations

In this article, we analyze a quadratic equal‐order stabilized finite element approximation for the incompressible Stokes equations based on two local Gauss integrations. Our method only offsets the discrete pressure gradient space by the residual of the simple and symmetry term at element level to circumvent the inf‐sup condition. And this method does not require specification of a stabilization parameter, and always leads to a symmetric linear system. Furthermore, this method is unconditionally stable, and can be implemented at the element level with minimal additional cost. Finally, we give some numerical simulations to show good stability and accuracy properties of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Residual a posteriori error estimates for a 3D mortar finite-element method: the Stokes system

We consider a non-conforming domain decomposition techniquefor the discretization of the three-dimensional Stokes equationsby the mortar finite-element method. Relying on the velocity–pressureformulation of the system, we perform the numerical analysisof residual error indicators for this problem and we prove thatthe error estimators provide upper and lower bounds for theenergy norm of the mortar finite-element solution.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A projection-based stabilized finite element method for steady-state natural convection problem

We formulate a projection-based stabilization finite element technique for solving steady-state natural convection problems. In particular, we consider heat transport through combined solid and fluid media. This stabilization does not act on the large flow structures. Based on the projection stabilization idea, finite element error analysis of the problem is investigated and optimal errors for the velocity, temperature and pressure are established. We also present some numerical tests which both verify the theoretical predictions and demonstrate the method?s promise.     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

A stabilized finite element method for a fictitious domain problem allowing small inclusions

The purpose of this work is to approximate numerically an elliptic partial differential equation posed on domains with small perforations (or inclusions). The approach is based on the fictitious domain method, and as the method's interest lies in the case in which the geometrical features are not resolved by the mesh, we propose a stabilized finite element method. The stabilization term is a simple, non‐consistent penalization that can be linked to the Barbosa‐Hughes approach. Stability and convergence are proved, and numerical results confirm the theory.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

A posteriori error estimates for a compressible Stokes system

A linearized compressible viscous Stokes system is considered. The a posteriori error estimates are defined and compared with the true error. They are shown to be globally upper and locally lower bounds for the true error of the finite element solution. Some numerical examples are given, showing an efficiency of the estimator. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 412–431, 2004.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

A posteriori estimates for the Stokes eigenvalue problem

We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a‐posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A posteriori error analysis for solving the Navier‐Stokes problem and convection‐diffusion equation

In this article, we consider the finite element discretization of the Navier‐Stokes problem coupled with convection‐diffusion equations where both the viscosity and the diffusion coefficients depend on the temperature. Existence and uniqueness of a solution are established. We prove a posteriori error estimates.     

A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.