Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Optimal estimates on stabilized finite volume methods for the three dimensional Navier–Stokes model are investigated and developed in this paper. Based on the global existence theorem [23], we first prove the global bound for the velocity in the H¹‐norm in time of a solution for suitably small data, and uniqueness of a suitably small solution by contradiction. Then, a full set of estimates is then obtained by some classical Galerkin techniques based on the relationship between finite element methods and finite volume methods approximated by the lower order finite elements for the three dimensional Navier–Stokes model.     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

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.     

A high order finite volume method for one dimensional nonlocal reactive flows of parabolic type

In this paper, we consider a high order finite volume approximation of one‐dimensional nonlocal reactive flows of parabolic type. The method is obtained by discretizing in space by arbitrary order vertex‐centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher‐order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H¹‐norm and L²‐norm, which are also confirmed by numerical tests. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite volume element method for monotone nonlinear elliptic problems

In this article, we consider the finite volume element method for the monotone nonlinear second‐order elliptic boundary value problems. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz‐continuous, and with the minimal regularity assumption on the exact solution, that is, u∈H¹(Ω), we show that the finite volume element method has a unique solution, and the finite volume element approximation is uniformly convergent with respect to the H¹ ‐norm. If u∈H^1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate \begin{align*}\mathcal{O}(h^{\epsilon})\end{align*} in the H¹ ‐norm. Moreover, we propose a natural and computationally easy residual‐based H¹ ‐norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Finite volume element approximations of nonlocal reactive flows in porous media

In this article, we study finite volume element approximations for two‐dimensional parabolic integro‐differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher‐order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H¹‐ and L²‐norm and are superconvergent in a discrete H¹‐norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L^∞‐ and W^1,∞‐norms. These results are also new for finite volume element methods for elliptic and parabolic equations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 285–311, 2000     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.     

A finite volume method for solving Navier-Stokes problems

We develop a finite volume method for solving the Navier-Stokes equations on a triangular mesh. We prove that the unique solution of the finite volume method converges to the true solution with optimal order for velocity and for pressure in discrete H¹ norm and L² norm respectively.     

Analysis and convergence of a MAC-like scheme for the generalized Stokes problem

We introduce a MAC-like scheme (a covolume method on rectangular grids) for approximating the generalized Stokes problem on an axiparallel domain. Two staggered grids are used in the derivation of the discretization. The velocity is approximated by conforming bilinears over rectangular elements, and the pressure by piecewise constants over macro-rectangular elements. The error in the velocity in the H¹ norm and the pressure in the L² norm are shown to be of first order, provided that the exact velocity is in H² and the exact pressure in H¹, and that the partition family of the domain is regular. © 1997 John Wiley & Sons, Inc.     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P ₁–P ₁ pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H ¹-norm for velocity and an estimate in the L ²-norm for pressure are obtained. An optimal error estimate in the L ²-norm for the velocity is derived under an additional assumption on the body force. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation.     

A posteriori error estimates for mixed finite element approximations of parabolic problems

We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L ²(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.     

Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint

In this paper, we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite elements and the control is discretized by piecewise constant functions. We prove the superconvergence error estimate of h² in L²-norm between the approximated solution and the interpolation of the exact control variable. Moreover, by postprocessing technique, we find that the projection of the discrete adjoint state is superclose (in order h²) to the exact control variable.     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.     

Stabilized DDFV schemes for stokes problem with variable viscosity on general 2D meshes

“Discrete Duality Finite Volume” schemes (DDFV for short) on general meshes are studied here for Stokes problems with variable viscosity with Dirichlet boundary conditions. The aim of this work is to analyze the well‐posedness of the scheme and its convergence properties. The DDFV method requires a staggered scheme, the discrete unknowns, the components of the velocity and the pressure, are located on different nodes. The scheme is stabilized using a finite volume analogue to Brezzi‐Pitkäranta techniques. This scheme is proved to be well‐posed on general meshes and to be first order convergent in a discrete H¹ ‐norm and a discrete L² ‐norm for respectively the velocity and the pressure. Finally, numerical experiments confirm the theoretical prediction, in particular on locally refined non conformal meshes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1666–1706, 2011     

An adaptive algorithm for the Crank–Nicolson scheme applied to a time-dependent convection–diffusion problem

An a posteriori upper bound is derived for the nonstationary convection–diffusion problem using the Crank–Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.A space and time adaptive algorithm is developed to ensure the control of the relative error in the L²(H¹) norm. Numerical experiments illustrating the efficiency of this approach are reported; it is shown that the error indicator is of optimal order with respect to both the mesh size and the time step, even in the convection dominated regime and in the presence of boundary layers.     

Approximation of the solution of a fourth order boundary value problem with nonsmooth coefficient

Summary A class of generalized finite element methods for the approximate solution of fourth order two point boundary value problem with nonsmooth coefficient is presented. The methods are based on the use of problem dependentL-splines incorporating the nonsmoothness of the coefficient. Stability is proved and optimal error estimates in theH ² norm are derived for the solution and postprocessed solution, under the assumption that the coefficient is of bounded variation. The relation of these methods to mixed methods is discussed.This research was sponsored by the Senate Research Committee of Syracuse University, Syracuse, NY 13210     

Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners

In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C¹-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2^nd order and its Laplacian are approximated simultaneously by C⁰-finite elements being piecewise polynomials.