Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.     

Discontinuous Galerkin methods for solving a hyperbolic inequality

In this paper, we study spatially semi‐discrete and fully discrete schemes to numerically solve a hyperbolic variational inequality, with discontinuous Galerkin (DG) discretization in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for four DG schemes, which reaches the optimal convergence order for linear elements. A numerical example is presented, and the numerical results confirm the theoretical error estimates.     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds

We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L²‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

In this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.     

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     

DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR A FORWARD-BACKWARD HEAT EQUATION

A space-time finite element method,discontinuous in time but continuous in space, is studied to solve the nonlinear forward-backward heat equation. A linearized technique is introduced in order to obtain the error estimates of the approximate solutions. And the numerical simulations are given.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

In this paper, we discuss the numerical simulation for a class of constrained optimal control problems governed by integral equations. The Galerkin method is used for the approximation of the problem. A priori error estimates and a superconvergence analysis for the approximation scheme are presented. Based on the results of the superconvergence analysis, a recovery type a posteriori error estimator is provided, which can be used for adaptive mesh refinement. The research project is supported by the National Basic Research Program under the Grant 2005CB321701 and the National Natural Science Foundation of China under the Grant 10771211.     

A posteriori error estimates for discontinuous Galerkin methods of obstacle problems

We present a posteriori error analysis of discontinuous Galerkin methods for solving the obstacle problem, which is a representative elliptic variational inequality of the first kind. We derive reliable error estimators of the residual type. Efficiency of the estimators is theoretically explored and numerically confirmed.     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non‐symmetric interior penalty discontinuous Galerkin finite element method, and a displacement‐velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

Convergence analysis of a modified weak Galerkin finite element method for Signorini and obstacle problems

In this article, we apply a modified weak Galerkin method to solve variational inequality of the first kind which includes Signorini and obstacle problems. Optimal order a priori error estimates in the energy norm are derived. We also provide some numerical experiments to validate the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1459–1474, 2017     

Superconvergence estimates of finite element methods for American options

In this paper we are concerned with finite element approximations to the evaluation of American options. First, following W. Allegretto etc., SIAM J. Numer. Anal. 39 (2001), 834–857, we introduce a novel practical approach to the discussed problem, which involves the exact reformulation of the original problem and the implementation of the numerical solution over a very small region so that this algorithm is very rapid and highly accurate. Secondly by means of a superapproximation and interpolation postprocessing analysis technique, we present sharp L ²-, L ^∞-norm error estimates and an H ¹-norm superconvergence estimate for this finite element method. As a by-product, the global superconvergence result can be used to generate an efficient a posteriori error estimator. This work was supported in part by the National Natural Science Foundation of China (10471103 and 10771158), the National Basic Research Program (2007CB814906), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), Tianjin Natural Science Foundation (07JCY-BJC14300), and Tianjin University of Finance and Economics.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.