Superconvergence of finite volume element method for a nonlinear elliptic problem

In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H¹ and W^1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the L^p and W^1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A posterior error estimates for the nonlinear grating problem with transparent boundary condition

The nonlinear grating problem is modeled by Maxwell's equations with transparent boundary conditions. The nonlocal boundary operators are truncated by taking sufficiently many terms in the corresponding expansions. A finite element method with the truncation operators is developed for solving the nonlinear grating problem. The two posterior error estimates are established. The a posterior error estimate consists of two parts: finite element discretization error and the truncation error of the nonlocal boundary operators. In particular, the truncation error caused by truncation operations is exponentially decayed when the parameter N is increased. Numerical experiment is included to illustrate the efficiency of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1101–1118, 2015     

On a mixed finite element formulation of a second‐order quasilinear problem in the plane

We analyze a mixed finite element discretization of a second‐order quasilinear problem based on the Raviart‐Thomas space. We prove that the discrete problem is solvable and provide a local uniqueness result for the solution. We also obtain optimal order L²‐error estimates for both the scalar variable and the associated flux. The main feature of our method is that it is free from the boundness conditions required in previous works on the coefficients of the quasilinear operator. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 90–103, 2004.     

Uniform superconvergence of a Galerkin finite element method on Shishkin‐type meshes

We consider a Galerkin finite element method that uses piecewise bilinears on a class of Shishkin‐type meshes for a model singularly perturbed convection‐diffusion problem on the unit square. The method is shown to be convergent, uniformly in the diffusion parameter ϵ, of almost second order in a discrete weighted energy norm. As a corollary, we derive global L₂‐norm error estimates and local L_∞‐norm estimates. Numerical experiments support our theoretical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16:426–440, 2000     

Superconvergent error estimates for a class of discretization methods for a coupled first‐order system with discontinuous coefficients

Lithological discontinuities in a reservoir generate discontinuous coefficients for the first‐order system of equations used in the simulation of fluid flow in porous media. Systems of conservation laws with discontinuous coefficients also arise in many other physical applications. In this article, we present a class of discretization schemes that include variants of mixed finite element methods, finite volume element methods, and cell‐centered finite difference equations as special cases. Error estimates of the order O(h²) in certain discrete L²‐norms are established for both the primary independent variable and its flux, even in the presence of discontinuous coefficients in the flux term. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 267–283, 1999     

Negative norm least‐squares methods for the velocity‐vorticity‐pressure Navier–Stokes equations

We develop and analyze a least‐squares finite element method for the steady state, incompressible Navier–Stokes equations, written as a first‐order system involving vorticity as new dependent variable. In contrast to standard L² least‐squares methods for this system, our approach utilizes discrete negative norms in the least‐squares functional. This allows us to devise efficient preconditioners for the discrete equations, and to establish optimal error estimates under relaxed regularity assumptions. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 237–256, 1999     

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

In this article, a finite element scheme based on the Newton's method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank‐Nicolson method is used for time discretization. Well‐posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

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     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

Finite element approximations with quadrature for second‐order hyperbolic equations

In this article, the effect of numerical quadrature on the finite element Galerkin approximations to the solution of hyperbolic equations has been studied. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in the L^∞(H¹), L^∞(L²) norms, whereas quasi‐optimal estimate is derived in the L^∞(L^∞) norm using energy methods. The analysis in the present paper improves upon the earlier results of Baker and Dougalis [SIAM J Numer Anal 13 (1976), pp 577–598] under the minimum smoothness assumptions of Rauch [SIAM J Numer Anal 22 (1985), pp 245–249] for a purely second‐order hyperbolic equation with quadrature. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 537–559, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10022     

A coupling of nonconforming and mixed finite element methods for Biot's consolidation model

In this article, we develop a nonconforming mixed finite element method to solve Biot's consolidation model. In particular, this work has been motivated to overcome nonphysical oscillations in the pressure variable, which is known as locking in poroelasticity. The method is based on a coupling of a nonconforming finite element method for the displacement of the solid phase with a standard mixed finite element method for the pressure and velocity of the fluid phase. The discrete Korn's inequality has been achieved by adding a jump term to the discrete variational formulation. We prove a rigorous proof of a‐priori error estimates for both semidiscrete and fully‐discrete schemes. Optimal error estimates have been derived. In particular, optimality in the pressure, measured in different norms, has been proved for both cases when the constrained specific storage coefficient c₀ is strictly positive and when c₀ is nonnegative. Numerical results illustrate the accuracy of the method and also show the effectiveness of the method to overcome the nonphysical pressure oscillations. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

The modified method of characteristics with mixed finite element domain decomposition procedures for the transient behavior of a semiconductor device

For the transient behavior of a semiconductor device, the modified method of characteristics with mixed finite element domain decomposition procedures applicable to parallel arithmetic is put forward. The electric potential equation is described by the mixed finite element method, and the electric, hole concentration and heat conduction equations are treated by the modified method of characteristics finite element domain decomposition methods. Some techniques, such as calculus of variations, domain decomposition, characteristic method, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in L² norm are derived for the error in the approximation solution. Thus the well‐known theoretical problem has been thoroughly and completely solved.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 353–368 2012     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Lp error estimates and superconvergence for covolume or finite volume element methods

We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the L^p norm, 2 ≤ p ≤ ∞, are derived. We also show second‐order convergence in the L^p norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the “supercloseness” results in Chou and Li [Math Comp 69(229) (2000), 103–120] to the L^p based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463–486, 2003     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

The Euler-Galerkin finite element method for nonlocal diffusion problems with a p-Laplace-type operator

This paper is devoted to the study of the finite element method for a class of non-linear nonlocal diffusion problems associated with p-Laplace-type operator. Using the Euler–Galerkin finite element method, the convergence and a priori error estimates for the semi-discrete as well as fully-discrete formulations are established.     

Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation

This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L²(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.     

Finite element approximations to one‐phase nonlinear free boundary problem in groundwater contamination flow

In this article, we consider a single‐phase coupled nonlinear Stefan problem of the water‐head and concentration equations with nonlinear source and permeance terms and a Dirichlet boundary condition depending on the free‐boundary function. The problem is very important in subsurface contaminant transport and remediation, seawater intrusion and control, and many other applications. While a Landau type transformation is introduced to immobilize the free boundary, a transformation for the water‐head and concentration functions is defined to deal with the nonhomogeneous Dirichlet boundary condition, which depends on the free boundary function. An H¹‐finite element method for the problem is then proposed and analyzed. The existence of the approximation solution is established, and error estimates are obtained for both the semi‐discrete schemes and the fully discrete schemes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006