A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017     

A partially penalty immersed interface finite element method for anisotropic elliptic interface problems

In this article, we develop a partially penalty immersed interface finite element (PIFE) method for a kind of anisotropy diffusion models governed by the elliptic interface problems with discontinuous tensor‐coefficients. This method is based on linear immersed interface finite elements (IIFE) and applies the discontinuous Galerkin formulation around the interface. We add two penalty terms to the general IIFE formulation along the sides intersected with the interface. The flux jump condition is weakly enforced on the smooth interface. By proving that the piecewise linear function on an interface element is uniquely determined by its values at the three vertices under some conditions, we construct the finite element spaces. Therefore, a PIFE procedure is proposed, which is based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. Then we prove the consistency and the solvability of the procedure. Theoretical analysis and numerical experiments show that the PIFE solution possesses optimal‐order error estimates in the energy norm and norm.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1984–2028, 2014     

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L^∞(L²) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps

We study smoothing properties and approximation of time derivativesfor time discretization schemes with constant time steps fora homogeneous parabolic problem formulated as an abstract initial-valueproblem in a Banach space. The time stepping schemes are basedon using rational functions r(z) e^–z which are A()-stablefor suitable [0, /2] and satisfy |r()| < 1, and the approximationsof time derivatives are based on using difference quotientsin time. Both smooth and non-smooth data error estimates ofoptimal order for the approximation of time derivatives areproved. Further, we apply the results to obtain error estimatesof time derivatives in the supremum norm for fully discretemethods based on discretizing the spatial variable by a finite-elementmethod.     

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     

A discontinuous finite volume element method for second‐order elliptic problems

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     

Error estimates for finite volume element methods for general second‐order elliptic problems

We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H¹ error estimate, H¹ superconvergence and L^p (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003.     

Partially penalized immersed finite element methods for parabolic interface problems

We present partially penalized immersed finite element methods for solving parabolic interface problems on Cartesian meshes. Typical semidiscrete and fully discrete schemes are discussed. Error estimates in an energy norm are derived. Numerical examples are provided to support theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1925–1947, 2015     

Discontinuous finite volume element method for parabolic problems

In this article, we consider the semidiscrete and the backward Euler fully discrete discontinuous finite volume element methods for the second‐order parabolic problems and obtain the optimal order error estimates in a mesh dependent norm and in the L²‐norm. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the refined maximum principle for degenerate elliptic and parabolic problems

We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems.     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.     

High‐order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007     

An unfitted interface penalty method for the numerical approximation of contrast problems

We aim to approximate contrast problems by means of a numerical scheme which does not require that the computational mesh conforms with the discontinuity between coefficients. We focus on the approximation of diffusion-reaction equations in the framework of finite elements. In order to improve the unsatisfactory behavior of Lagrangian elements for this particular problem, we resort to an enriched approximation space, which involves elements cut by the interface. Firstly, we analyze the H¹-stability of the finite element space with respect to the position of the interface. This analysis, applied to the conditioning of the discrete system of equations, shows that the scheme may be ill posed for some configurations of the interface. Secondly, we propose a stabilization strategy, based on a scaling technique, which restores the standard properties of a Lagrangian finite element space and results to be very easily implemented. We also address the behavior of the scheme with respect to large contrast problems ending up with a choice of Nitsche?s penalty terms such that the extended finite element scheme with penalty is robust for the worst case among small sub-elements and large contrast problems. The theoretical results are finally illustrated by means of numerical experiments.     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

Higher‐order finite volume element methods based on Barlow points for one‐dimensional elliptic and parabolic problems

The article is devoted to a kind of higher‐order finite volume element methods, where the dual partitions are constructed by Barlow points, for elliptic and parabolic problems in one space dimension. Techniques to derive the stability and to control the nonsymmetry are presented. Superconvergence and the optimal order errors in the H¹‐ and L²‐norms are obtained. Numerical results illustrate the theoretical findings. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 977–994, 2015