A mortar mimetic finite difference method on non-matching grids

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results. Supported by the US Department of Energy, under contractW-7405-ENG-36. LA-UR-04-4740. Partially supported by NSF grants EIA 0121523 and DMS 0411413, by NPACI grant UCSD 10181410, and by DOE grant DE-FGO2-04ER25617. Partially supported by NSF grants DMS 0107389, DMS 0112239 and DMS 0411694 and by DOE grant DE-FG02-04ER25618.     

Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations. Received November 2, 1995 / Revised version received February 10, 1997     

A finite difference method for free boundary problems

Fornberg and Meyer-Spasche proposed some time ago a simple strategy to correct finite difference schemes in the presence of a free boundary that cuts across a Cartesian grid. We show here how this procedure can be combined with a minimax-based optimization procedure to rapidly solve a wide range of elliptic-type free boundary value problems.     

Application of the generalized finite difference method to solve the advection-diffusion equation

The study of the advection-diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering.This paper shows the application of the generalized finite difference method to solve the advection-diffusion equation by the explicit method. The convergence of the method has been studied and the truncation error over irregular grids is given. An example has been solved using the explicit finite difference formulae and the criterion of stability.     

Superconvergent biquadratic finite volume element method for two-dimensional Poisson’s equations

In this paper, a kind of biquadratic finite volume element method is presented for two-dimensional Poisson’s equations by restricting the optimal stress points of biquadratic interpolation as the vertices of control volumes. The method can be effectively implemented by alternating direction technique. It is proved that the method has optimal energy norm error estimates. The superconvergence of numerical gradients at optimal stress points is discussed and it is proved that the method has also superconvergence displacement at nodal points by a modified dual argument technique. Finally, a numerical example verifies the theoretical results and illustrates the effectiveness of the method.     

A new TVD flux-limiter method for solving nonlinear hyperbolic equations

We develop a new flux-limiter method based on the Richtmyer two-step Lax-Wendroff (R2LW) method coupled with a conservative upwind method and a nonconventional flux-limiter function. The proposed method is TVD stable and preserves the linear stability condition of the R2LW method. The numerical results show that this method improves on the approximations obtained by standard first-order and second-order TVD schemes. The new method provides accurate results for nonlinear hyperbolic equations with discontinuous solutions.     

Solving parabolic and hyperbolic equations by the generalized finite difference method

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points.     

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations

This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.     

The role of coefficients of a general SPDE on the stability and convergence of a finite difference method

In this paper for the approximate solution of stochastic partial differential equations (SPDEs) of Itô-type, the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed. The finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported. We prove the consistency and stability of these methods and investigate the influence of the multiplier (particularly multiplier of the random noise) in mean square stability. From stochastic version of Lax-Richtmyer the convergence of these methods under some conditions are established. Numerical experiments are included to show the efficiency of the methods.     

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.     

Maximum norm error estimates of efficient difference schemes for second-order wave equations

The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results.     

A robust and accurate finite difference method for a generalized Black-Scholes equation

In this paper we present a numerical method for a generalized Black-Scholes equation, which is used for option pricing. The method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. Our scheme is stable for arbitrary volatility and arbitrary interest rate, and is second-order convergent with respect to the spatial variable. Furthermore, the present paper efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients

In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented.     

Monotone iterative methods for finite difference system of reaction-diffusion equations

Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.     

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H¹-norm is proved to be O(h+H³|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.     

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

A uniformly convergent scheme for a system of reaction–diffusion equations

In this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically.