OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

Analysis of a finite volume element method for the Stokes problem

In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries. Finally, some numerical experiments that confirm the predicted behavior of the method are provided.     

Generalizations of the Finite Element Method

This paper is concerned with the generalization of the finite element method via the use of non-polynomial enrichment functions. Several methods employ this general approach, e.g. the extended finite element method and the generalized finite element method. We review these approaches and interpret them in the more general framework of the partition of unity method. Here we focus on fundamental construction principles, approximation properties and stability of the respective numerical method. To this end, we consider meshbased and meshfree generalizations of the finite element method and the use of smooth, discontinuous, singular and numerical enrichment functions.     

Finite volume element approximation and analysis for a kind of semiconductor device simulation

In this paper, we present a finite volume element scheme for a kind of two dimensional semiconductor device simulation. A general framework is developed for finite volume element approximation of the semiconductor problems. We construct a fully discrete finite volume element scheme based on triangulations with a piecewise linear finite element space and a general type of control volume. Optimal-order convergence in H ¹-norm is derived.     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.     

A BALANCED OVERSAMPLING FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS WITH OBSERVATIONAL BOUNDARY DATA

In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz $\psi_2$-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.     

First-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods

In this paper we show first-order convergence of a multi-point flux approximation control volume method (MPFA) on unstructured triangular grids. In this approach the flux approximation is derived directly in the physical space. In order to do this, we introduce a perturbed mixed finite element method that is equivalent to the MPFA scheme and prove the first-order convergence of this approach. Moreover, we carefully compare the computational performance properties of the MPFA method with those of a lowest order Raviart–Thomas and Brezzi–Douglas–Marini mixed finite element approximation.     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.     

A finite volume method for scalar conservation laws with stochastic time–space dependent flux functions

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions.     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

对流扩散方程的有限体积-有限元方法的误差估计

本文结合有限体积方法和有限元方法处理非线性对流扩散问题,非线性对流项利用有限体积方法处理,扩散项利用有限元方法离散,并给近似解的误差估计。     

Convergence and quasi-optimality of adaptive nonconforming finite element methods for some nonsymmetric and indefinite problems

Recently an adaptive nonconforming finite element method (ANFEM) has been developed by Carstensen and Hoppe (in Numer Math 103:251–266, 2006). In this paper, we extend the result to some nonsymmetric and indefinite problems. The main tools in our analysis are a posteriori error estimators and a quasi-orthogonality property. In this case, we need to overcome two main difficulties: one stems from the nonconformity of the finite element space, the other is how to handle the effect of a nonsymmetric and indefinite bilinear form. An appropriate adaptive nonconforming finite element method featuring a marking strategy based on the comparison of the a posteriori error estimator and a volume term is proposed for the lowest order Crouzeix–Raviart element. It is shown that the ANFEM is a contraction for the sum of the energy error and a scaled volume term between two consecutive adaptive loops. Moreover, quasi-optimality in the sense of quasi-optimal algorithmic complexity can be shown for the ANFEM. The results of numerical experiments confirm the theoretical findings.     

A mixed finite element for the stokes problem using quadrilateral elements

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.     

A mixed finite element for the stokes problem using quadrilateral elements

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.     

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems

The finite volume element method is a discretization technique for partial differential equations, but in general case the coefficient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modified finite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coefficient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term.     

Meshfree First-order System Least Squares

We prove convergence for a meshfree first-order system least squares （FOSLS） partition of unity finite element method （PUFEM）. Essentially, by virtue of the partition of unity, local approximation gives rise to global approximation in H（div）∩H（curl）. The FOSLS formulation yields local a posteriori error estimates to guide the judicious allotment of new degrees of freedom to enrich the initial point set in a meshfree discretization. Preliminary numerical results are provided and remaining challenges are discussed.