A POSTERIORI ERROR ESTIMATES OF A NON-CONFORMING FINITE ELEMENT METHOD FOR PROBLEMS WITH ARTIFICIAL BOUNDARY CONDITIONS

An a posteriori error estimator is obtained for a nonconforming finite element approximation of a linear elliptic problem, which is derived from a corresponding unbounded domain problem by applying a nonlocal approximate artificial boundary condition. Our method can be easily extended to obtain a class of a posteriori error estimators for various conforming and nonconforming finite element approximations of problems with different artificial boundary conditions. The reliability and efficiency of our a posteriori error estimator are rigorously proved and are verified by numerical examples.     

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO- AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.     

A LOCAL MULTILEVEL PRECONDITIONER FOR THE ADAPTIVE MORTAR FINITE ELEMENT METHOD＊

In this paper, we propose a local multilevel preconditioner for the mortar finite element approximations of the elliptic problems. With some mesh assumptions on the interface, we prove that the condition number of the preconditioned systems is independent of the large jump of the coefficients but depends on the mesh levels around the cross points. Some numericM experiments are presented to confirm our theoreticM results.     

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.     

ADAPTIVE FINITE ELEMENT APPROXIMATION FOR A CLASS OF PARAMETER ESTIMATION PROBLEMS

In this paper, we study adaptive finite element discretisation schemes for a class of parameter estimation problem. We propose to efficient algorithms for the estimation problem use adaptive multi-meshes in developing We derive equivalent a posteriori error estimators for both the state and the control approximation, which particularly suit an adaptive multi-mesh finite element scheme. The error estimators are then implemented and tested with promising numerical results.     

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL GOVERNED BY CONVECTION DOMINATED DIFFUSION EQUATIONS

In this paper, we study variational discretization for the constrained optimal control problem governed by convection dominated diffusion equations, where the state equation is approximated by the edge stabilization Galerkin method. A priori error estimates are derived for the state, the adjoint state and the control. Moreover, residual type a posteriori error estimates in the L^2-norm are obtained. Finally, two numerical experiments are presented to illustrate the theoretical results.     

SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS

We propose and analyze a C^0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is established for a simplified model.     

A POSTERIORI ERROR ESTIMATES FOR LOCAL CO DISCONTINUOUS GALERKIN METHODS FOR KIRCHHOFF PLATE BENDING PROBLEMS

We derive some residual-type a posteriori error estimates for the local CO discontinuous Galerkin （LCDG） approximations （[31]） of the Kirchhoff bending plate clamped on the boundary. The estimator is both reliable and efficient with respect to the moment-field approximation error in an energy norm. Some numerical experiments are reported to demonstrate theoretical results.     

EFFICIENT NUMERICAL ALGORITHMS FOR THREE-DIMENSIONAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

This paper detailedly discusses the locally one-dimensional numerical methods for ef- ficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional dif- fusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.     

A MIXED FINITE ELEMENT METHOD ON A STAGGERED MESH FOR NAVIER-STOKES EQUATIONS

In this paper, we introduce a mixed finite element method on a staggered mesh for the numerical solution of the steady state Navier-Stokes equations in which the two components of the velocity and the pressure are defined on three different meshes. This method is a conforming quadrilateral Q1 × Q1 - P0 element approximation for the Navier-Stokes equations. First-order error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to illustrate the effectiveness of the proposed method.     

A COARSENING ALGORITHM ON ADAPTIVE GRIDS BY NEWEST VERTEX BISECTION AND ITS APPLICATIONS

In this paper, an efficient and easy-to-implement coarsening algorithm is proposed for adaptive grids obtained using the newest vertex bisection method in two dimemsions. The new coarsening algorithm does not require storing the binary refinement tree explicitly. Instead, the structure is implicitly contained in a special ordering of triangular elements. Numerical experiments demonstrate that the proposed coarsening algorithm is efficient when applied for multilevel preconditioners and mesh adaptivity for time-dependent problems.     

THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.     

COUPLING OF FINITE ELEMENT AND BOUNDARY ELEMENT METHODS FOR THE SCATTERING BY PERIODIC CHIRAL STRUCTURES

Consider a time-harmonic electromagnetic plane wave incident on a biperiodic structure in R^3. The periodic structure separates two homogeneous regions. The medium inside the structure is chiral and nonhomogeneous. In this paper, variational formulations coupling finite element methods in the chiral medium with a method of integral equations on the periodic interfaces are studied. The well-posedness of the continuous and discretized problems is established. Uniform convergence for the coupling variational approximations of the model problem is obtained.     

UNIFORMLY A POSTERIORI ERROR ESTIMATE FOR THE FINITE ELEMENT METHOD TO A MODEL PARAMETER DEPENDENT PROBLEM

This paper proposes a reliable and efficient a posteriori error estimator for the finite element methods for the beam problem. It is proved that the error can be bounded by the computable error estimator from above and below up to multiplicative constants that do neither depend on the meshsize nor on the thickness of the beam.     

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.     

SUPERCONVERGENCE OF GRADIENT RECOVERY SCHEMES ON GRADED MESHES FOR CORNER SINGULARITIES

For the linear finite element solution to the Poisson equation, we show that supercon- vergence exists for a type of graded meshes for corner singularities in polygonal domains. In particular, we prove that the L^2-projection from the piecewise constant field △↓UN to the continuous and piecewise linear finite element space gives a better approximation of △↓U in the Hi-norm. In contrast to the existing superconvergence results, we do not assume high regularity of the exact solution.     

Model order reduction and error estimation with an application to the parameter-dependent eddy current equation

In product development, engineers simulate the underlying partial differential equation many times with commercial tools for different geometries. Since the available computation time is limited, we look for reduced models with an error estimator that guarantees the accuracy of the reduced model. Using commercial tools the theoretical methods proposed by G. Rozza, D.B.P. Huynh and A.T. Patera [Reduced basis approximation and a posteriori error estimation for affinely parameterized elliptic coercive partial differential equations, Arch. Comput. Methods Eng. 15 (2008), pp. 229–275] lead to technical difficulties. We present how to overcome these challenges and validate the error estimator by applying it to a simple model of a solenoid actuator that is a part of a valve.     

HOMOTOPY CONTINUATION METHODS FOR STOCHASTIC TWO-POINT BOUNDARY VALUE PROBLEMS DRIVEN BY ADDITIVE NOISES

In this paper the homotopy continuation method for stochastic two-point boundary value problems driven by additive noises is studied. The existence of the solution of the homotopy equation is proved. Numerical schemes are constructed and error estimates are obtained. Numerical experiments demonstrate the effectiveness of the homotopy continu- ation method over other commonly used methods such as the shooting method.     

THE ELLIPSOID ARTIFICIAL BOUNDARY METHOD FOR THREE-DIMENSIONAL UNBOUNDED DOMAINS

The artificial boundary method is applied to solve three-dimensional exterior problems. Two kind of rotating ellipsoids are chosen as the artificial boundaries and the exact artificial boundary conditions are derived explicitly in terms of an infinite series. Then the well-posedness of the coupled variational problem is obtained. It is found that error estimates derived depend on the mesh size, truncation term and the location of the artificial boundary. Three numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.