AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION

In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is de- duced using Benssoussau-Lions＇ algorithm. Finally, the results of some numerical experiments are presented to support the theory.

RESIDUAL A POSTERIORI ERROR ESTIMATE TWO-GRID METHODS FOR THE STEADY (NAVIER-STOKES) EQUATION WITH STREAM FUNCTION FORM

Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.

Residual-type a posteriori error estimate for parabolic obstacle problems

In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.

A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear,Dispersive Equations

The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

THE A POSTERIORI ERROR ESTIMATOR OF SDG METHOD FOR VARIABLE COEFFICIENTS TIME-HARMONIC MAXWELL'S EQUATIONS

In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations.We propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type estimator.We first propose the curl-recovery method for the staggered discontinuous Galerkin method(SDGM),and based on the super-convergence result of the postprocessed solution,an asymptotically exact error estimator is constructed.The residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations.The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.

A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques

Based on the auxiliary subspace techniques,a posteriori error estimator of nonconforming weak Galerkin finite element method(WGFEM)for Stokes problem in two and three dimensions is presented.Without saturation assumption,we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term.The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix.To reduce the computational cost of the error problem,an equivalent error problem is constructed by using diagonalization techniques,which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom(d.o.f)to get the error estimator.Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

High accuracy eigensolution and its extrapolation for potential equations

From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are presented to obtain the eigensolutions that are used to solve Laplace＇s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone＇s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm （EA）, the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.

Adaptive Finite Element Method for Steady Convection-Diffusion Equation

This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.

OPTIMAL A POSTERIORI ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD FOR CONVECTION- DIFFUSION PROBLEMS IN ONE SPACE DIMENSION

In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver- gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 （2015）, pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to （p ＋ 1）-degree right and left Radau interpolating polynomials under mesh re- finement. The order of convergence is proved to be p ＋ 2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to （p ＋ 1）-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at （.9（hp＋2） rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at （9（h） rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p＋3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using PP polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.

Adaptivity and A Posteriori Error Control for Bifurcation Problems Ⅰ:The Bratu Problem

This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied.In particular,we study in detail the numerical approximation of the Bratu problem,based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method.A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual(DWR)approach.Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.

A UNIFIED A POSTERIORI ERROR ANALYSIS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF REACTIVE TRANSPORT EQUATIONS

Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Galerkin （SIPG）, and incomplete interior penalty Galerkin （IIPG）. A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution and given data, explicit estimators can be computed efficiently and directly, which can be used as error indicators for adaptation. Unlike in the reference [10], we obtain the error estimators in L^2 （L^2） norm by using duality techniques instead of in L^2（H^1） norm.

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

A Novel Error Analysis of Spectral Method for the Anomalous Subdiffusion Problems with Multi-term Time-fractional Derivative

This paper aims to extend a space-time spectral method to address the multi-term time-fractional subdiffusion equations with Caputo derivative. In this method, the Jacobi polynomials are adopted as the basis functions for temporal discretization and the Lagrangian polynomials are used for spatial discretization. An efficient spectral approximation of the weak solution is established. The main work is the demonstration of the well-posedness for the weak problem and the derivation of a posteriori error estimates for the spectral Galerkin approximation. Extensive numerical experiments are presented to perform the validity of a posteriori error estimators, which support our theoretical results.

A POSTERIORI ERROR ESTIMATE FOR BOUNDARY CONTROL PROBLEMS GOVERNED BY THE PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we discuss the a posteriori error estimate of the finite element approximation for the boundary control problems governed by the parabolic partial differential equations. Three different a posteriori error estimators are provided for the parabolic boundary control problems with the observations of the distributed state, the boundary state and the final state. It is proven that these estimators are reliable bounds of the finite element approximation errors, which can be used as the indicators of the mesh refinement in adaptive finite element methods.

ON RESIDUAL-BASED A POSTERIORI ERROR ESTIMATORS FOR LOWEST-ORDER RAVIART-THOMAS ELEMENT APPROXIMATION TO CONVECTION-DIFFUSION-REACTION EQUATIONS

A new technique of residual-type a posteriori error analysis is developed for the lowest- order Raviart-Thomas mixed finite element discretizations of convection-diffusion-reaction equations in two- or three-dimension. Both centered mixed scheme and upwind-weighted mixed scheme are considered. The a posteriori error estimators, derived for the stress variable error plus scalar displacement error in L_2-norm, can be directly computed with the solutions of the mixed schemes without any additional cost, and are proven to be reliable. Local efficiency dependent on local variations in coefficients is obtained without any saturation assumption, and holds from the cases where convection or reaction is not present to convection- or reaction-dominated problems. The main tools of the analysis are the postprocessed approximation of scalar displacement, abstract error estimates, and the property of modified Oswald interpolation. Numerical experiments are carried out to support our theoretical results and to show the competitive behavior of the proposed posteriori error estimates.

SUPERCONVERGENCE AND A POSTERIORI ERROR ESTIMATES FOR BOUNDARY CONTROL GOVERNED BY STOKES EQUATIONS

In this paper, the superconvergence results are derived for a class of boundary control problems governed by Stokes equations. We derive superconvergence results for both the control and the state approximation. Base on superconvergence results, we obtain asymptotically exact a posteriori error estimates.

A POSTERIORI ENERGY-NORM ERROR ESTIMATES FOR ADVECTION-DIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS

We propose and analyze a posteriori energy-norm error estimates for weighted interior penalty discontinuous Galerkin approximations of advection-diffusion-reaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms： a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a non-conforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfiirth on continuous finite elements, that the local lower error bounds can be written with constants involving a cut-off for the ratio of local mesh size to the reciprocal of the square root of the lowest local eignevalue of the diffusion tensor.

A Posteriori Error Estimates for Conservative Local Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

We construct and analyze conservative local discontinuous Galerkin(LDG)methods for the Generalized Korteweg-de-Vries equation.LDG methods are designed by writing the equation as a system and performing separate approximations to the spatial derivatives.The main focus is on the development of conservative methods which can preserve discrete versions of the first two invariants of the continuous solution,and a posteriori error estimates for a fully discrete approximation that is based on the idea of dispersive reconstruction.Numerical experiments are provided to verify the theoretical estimates.

A Posteriori Error Estimates of the Galerkin Spectral Methods for Space-Time Fractional Diffusion Equations

In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.

A POSTERIORI ERROR ESTIMATE OF OPTIMAL CONTROL PROBLEM OF PDE WITH INTEGRAL CONSTRAINT FOR STATE

In this paper, we study adaptive finite element discretization schemes for an optimal control problem governed by elliptic PDE with an integral constraint for the state. We derive the equivalent a posteriori error estimator for the finite element approximation, which particularly suits adaptive multi-meshes to capture different singularities of the control and the state. Numerical examples are presented to demonstrate the efficiency of a posteriori error estimator and to confirm the theoretical results.