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.     

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H⁴ = ?(w²) = ?(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.     

Numerical solution of a second biharmonic boundary value problem

The second boundary value problem for the biharmonic equation is equivalent to the Dirichlet problems for two Poisson equations. Several finite difference approximations are defined to solve these Dirichlet problems and discretization error estimates are obtained. It is shown that the splitting of the biharmonic equation produces a numerically efficient procedure.     

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.     

Highly accurate numerical solutions with repeated Richardson extrapolation for 2D laplace equation

A theoretical basis is presented for the repeated Richardson extrapolation (RRE) to reduce and estimate the discretization error of numerical solutions for heat conduction. An example application is described for the 2D Laplace equation using the finite difference method, a domain discretized with uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 8,193 × 8,193 nodes, a multigrid method, single, double and quadruple precisions and up to twelve Richardson extrapolations. It was found that: (1) RRE significantly reduces the discretization error (for example, from 2.25E-07 to 3.19E-32 with nine extrapolations and a 1,025 × 1,025 grid, yielding an order of accuracy of 19.1); (2) the Richardson error estimator works for numerical results obtained with RRE; (3) a higher reduction of the discretization error with RRE is achieved by using higher calculation precision, a larger number of extrapolations, a larger number of grids and correct error orders; and (4) to obtain a given value error, much less CPU time and RAM memory are required for the solution with RRE than without it.     

Finite Volume Methods for Multi-Symplectic PDES

We investigate the application of a cell-vertex finite volume discretization to multi-symplectic PDEs. The investigated discretization reduces to the Preissman box scheme when used on a rectangular grid. Concerning arbitrary quadrilateral grids, we show that only methods with parallelogram-like finite volume cells lead to a multi-symplectic discretization; i.e., to a method that preserves a discrete conservation law of symplecticity. One of the advantages of finite volume methods is that they can be easily adjusted to variable meshes. But, although the implementation of moving mesh finite volume methods for multi-symplectic PDEs is rather straightforward, the restriction to parallelogram-like cells implies that only meshes moving with a constant speed are multi-symplectic. To overcome this restriction, we suggest the implementation of reversible moving mesh methods based on a semi-Lagrangian approach. Numerical experiments are presented for a one dimensional dispersive shallow-water system.     

Crank-Nicolson拟合有限体积法定价机制转换下的欧式期权（英文）

This paper develops and analyses a novel numerical scheme to price European options under regime switching model which is governed by a system of partial differential equations(PDEs).To numerically solve these PDEs,we introduce a fitted finite volume method for the spatial discretization,coupled with the Crank-Nicolson time stepping scheme.We show that this scheme is consistent,stable and monotone,and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the new numerical method.     

Mortar Finite Volume Method with Adini Element for Biharmonic Problem

In this paper, we construct and analyse a mortar finite volume method for the discretization for the biharmonic problem in R2. This method is based on the mortar-type Adini nonconforming finite element spaces. The optimal order H2-seminorm error estimate between the exact solution and the mortar Adini finite volume solution of the biharmonic equation is established.     

Repeated Richardson extrapolation applied to the two-dimensional Laplace equation using triangular and square grids

The focus of this work is to verify the efficiency of the Repeated Richardson Extrapolation (RRE) to reduce the discretization error in a triangular grid and to compare the result to the one obtained for a square grid for the two-dimensional Laplace equation. Two different geometries were employed: the first one, a unitary square domain, was discretized into a square or triangular grid; and the second, a half square triangle, was discretized into a triangular grid. The methodology employed used the following conditions: the finite volume method, uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 16,777,216 nodes for the square domain and up to 2097,152 nodes for the half square triangle domain, multigrid method, double precision, up to eleven Richardson extrapolations for the first domain and up to ten Richardson extrapolations for the second domain. It was verified that (1) RRE is efficient in reducing the discretization error in a triangular grid, achieving an effective order of approximately 11 for all the variables of interest for the first geometry; (2) for the same number of nodes and with or without RRE, the discretization error is smaller in a square grid than in a triangular grid; and (3) the magnitude of the numerical error reduction depends on, among other factors, the variable of interest and the domain geometry.     

A local least-squares method for solving nonlinear partial differential equations of second order

In this paper a mesh-free method for the treatment of time-independent and time-dependent nonlinear PDEs of second order is presented. The basic idea of the discretization is a local least-squares approximation, similar to the moving least-squares approach in data approximation. However, in our approach the PDE is incorporated as an additional minimization constraint. The discretization leads to a fixed-point problem, which is solved by iteration. Because of the local nature of the method only small dimensional matrix inversions have to be done. The approximation error of the discretization—even on unstructured meshes—is comparable to respective versions of finite elements. As a by-product the method provides an a posteriori measure for the local approximation error. We discuss implementational aspects and present numerical simulations.     

A multilevel algorithm for the biharmonic problem

Summary A finite element discretization of the mixed variable formulation of the biharmonic problem is considered. A multilevel algorithm for the numerical solution of the discrete equations is described. Convergence is proved under the assumption ofH ³-regularity.     

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.     

A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.     

A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach

In this article, using coupled approach, we discuss fourth order finite difference approximation for the solution of two dimensional nonlinear biharmonic partial differential equations on a 9‐point compact stencil. The solutions of unknown variable and its Laplacian are obtained at each internal grid points. This discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions. We require only system of two equations to obtain the solution and its Laplacian. The proposed fourth order method is used to solve a set of test problems and produce high accuracy numerical solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

A priori error estimates for upwind finite volume schemes for two-dimensional linear convection diffusion problems

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R² and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h^1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.     

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.     

ARTIFICIAL BOUNDARY METHOD FOR THE THREE-DIMENSIONAL EXTERIOR PROBLEM OF ELASTICITY

The exact boundary condition on a spherical artificial boundary is derived for thethree-dimensional exterior problem of linear elasticity in this paper. After this bound-ary condition is imposed on the artificial boundary, a reduced problem only defined in abounded domain is obtained. A series of approximate problems with increasing accuracycan be derived if one truncates the series term in the variational formulation, which isequivalent to the reduced problem. An error estimate is presented to show how the errordepends on the finite element discretization and the accuracy of the approximate problem.In the end, a numerical example is given to demonstrate the performance of the proposedmethod.     

Robust multigrid preconditioners for the high‐contrast biharmonic plate equation

We study the high‐contrast biharmonic plate equation with Hsieh–Clough–Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11 :319–331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high‐contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high‐contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright © 2010 John Wiley & Sons, Ltd.     

Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet‐Raviart scheme

Using the technique of eigenvalue error expansion and the technique of integral identities, we derive higher order convergence rate of the eigenvalue approximation for the biharmonic eigenvalue problem based on the Ciarlet‐Raviart discretization. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005