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In this paper, we study the residual-based a posteriori error estimates of h p -discontinuous Galerkin finite element methods for strongly nonlinear elliptic boundary value problems. Computable upper and lower bounds on the error are derived in a natural mesh-dependent energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. Numerical experiments are also provided to illustrate the performance of the proposed estimators.

A two-grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time-dependent advection-dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine-grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse-grid space (with grid size H) and a linear system on the fine-grid space. An optimal error estimate in H1 -norm is obtained for the two-grid method. It shows that the two-grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H2). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

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∈H1+e(Ω),0 < e ≤ 1, we develop the optimal convergence rate 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

In this article, we propose a new discontinuous finite volume element (DFVE) method for the second-order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L2 -norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012

In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L 2 and H 1 norms for the standard finite volume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency of the procedure.

Two-grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method. The methods are based on one coarse-grid space and one fine-grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine-grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h=O(H3|lnH|). As a result, solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation.

In this paper, we consider a semi-discrete mortar finite volume element method for two-dimensional parabolic problems. This method is based on the mortar Crouzeix-Raviart non-conforming finite element space. It is proved that the mortar finite volume element approximations derived are convergent with the optimal order in the H^1- and L^2-norms. Numerical experiments are presented to illustrate the theoretical results.

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for non-selfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H 1-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption

In this paper, we consider the superconvergence of a mixed covolume method on the quasi-uniform triangular grids for the variable coefficient-matrix Poisson equations. The superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained. With these superconvergence estimates, we establish the superconvergence estimates and the L∞-error estimates for the mixed covolume method for the elliptic problems. Based on the superconvergence of the mixed covolume method, under the condition that the triangulation is uniform, we construct a post-processing method for the approximate velocity which improves the order of approximation of the approximate velocity.

In this paper, we consider the finite volume element method based on the Crouzeix–Raviart element and prove the existence, uniqueness and uniform convergence of the finite volume element approximations for the non-self-adjoint and indefinite elliptic problems under minimal elliptic regularity assumption.

In this article, we consider the finite volume element method for the second-order nonlinear elliptic problem and obtain the H1 and W1,∞ 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 Lp and W1,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

We construct and analyse a mortar finite volume method for the discretization for selfadjoint elliptic boundary value problems in R 2 . This method is based on the mortar Crouzeix–Raviart non-conforming finite element spaces. We prove the optimal order H 1 -norm and L 2 -norm error estimates between the exact solution and the mortar finite volume approximation of the elliptic problems.

A least-squares mixed finite element method for the second-order non-self-adjoint two-point boundary value problems is formulated and analysed. Superconvergence estimates are developed in the maximum norm at Gaussian points and at Lobatto points.