Chebyshev tau meshless method based on the highest derivative for fourth order equations

It is well known that the numerical integration process is much less sensitive than numerical differential process when dealing with the differential equations. After integration, accuracy is no longer limited by that of the slowly convergent series for the highest derivative, but only by that of the unknown function itself. In this paper, a Chebyshev tau meshless method based on the highest derivative (CTMMHD) is developed for fourth order equations on irregularly shaped domains with complex boundary conditions. The problem domain is embedded in a domain of regular shape. The integration and multiplication of Chebyshev expansions are given in matrix representations. Several numerical experiments including standard biharmonic problems, problems with variable coefficients and nonlinear problems are implemented to verify the high accuracy and efficiency of our method.     

Approximate solution of boundary integral equations for biharmonic problems in non-smooth domains

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.     

Multigrid technique for biharmonic interpolation with application to dual and multiple reciprocity method

A biharmonic-type interpolation method is presented to solve 2D and 3D scattered data interpolation problems. Unlike the methods based on radial basis functions, which produce a large linear system of equations with fully populated and often non-selfadjoint and ill-conditioned matrix, the presented method converts the interpolation problem to the solution of the biharmonic equation supplied with some non-usual boundary conditions at the interpolation points. To solve the biharmonic equation, fast multigrid techniques can be applied which are based on a non-uniform, non-equidistant but Cartesian grid generated by the quadtree/octtree algorithm. The biharmonic interpolation technique is applied to the multiple and dual reciprocity method of the BEM to convert domain integrals to the boundary. This makes it possible to significantly reduce the computational cost of the evaluation of the appearing domain integrals as well as the memory requirement of the procedure. The resulting method can be considered as a special grid-free technique, since it requires no domain discretisation. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Exact Results in the Finite Element Method

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u. We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.     

On the boundary value problem of the biharmonic operator on domains with angular corners

The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.     

Difference methods of order two and four for systems of mildly nonlinear biharmonic problems of the second kind in two space dimensions

In this article, we report two sets of finite difference methods of order two and four over a rectangular domain for the efficient numerical integration of the system of two-dimensional nonlinear elliptic biharmonic problems of the second kind. Second-order derivatives of the solutions are obtained as byproducts of the methods. We use only 9 grid points and do not require fictitious points in order to approximate the boundary conditions. In numerical experiments, the new second- and fourth-order formulas are compared with the exact solutions both in singular and nonsingular cases. © 1996 John Wiley & Sons, Inc.     

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.     

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     

A consistency analysis for the numerical solution of boundary integral equations

A method is presented for assessing the nature of the error incurred in the boundary integral equation (BIE) solution of both harmonic and biharmonic boundary value problems (BVPs). It is shown to what order of accuracy the governing partial differential equation is actually represented by the approximating numerical scheme, and how raising the order of the boundary ‘shape functions’ affects this representation. The effect of varying both the magnitude and the aspect ratio of the solution domain is investigated; it is found that the present technique may suggest an optimum nondimensional scaling for the BIE solution of a particular harmonic or biharmonic BVP.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique

This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method.     

Construction of Bézier Surfaces from Prescribed Boundary

In this paper, we present a method for generating Bézier surfaces from the boundary information based on a general second order functional and a third order functional associated with the triharmonic equation. By solving simple linear equations, the internal control points of the resulting Bézier surface can be obtained as linear combinations of the given boundary control points. This is a generalization of previous works on Plateau-Bézier problem, harmonic, biharmonic and quasi-harmonic Bézier surfaces. Some representative examples show the effectiveness of the presented method.     

The Poisson and the Biharmonic Equations in the Interior of a Convex Polygon

A new method for analyzing linear elliptic partial differential equations in the interior of a convex polygon was developed in the late 1990s. This method does not rely on the classical approach of separation of variables and on the use of classical integral transforms and therefore is well suited for the investigation of the biharmonic equation. Here, we present a novel integral representation of the solution of the biharmonic equation in the interior of a convex polygon. This representation contains certain free parameters and therefore is more general than the one presented in [1]. For a given boundary value problem, by choosing these free parameters appropriately, one can obtain the simplest possible representation for the solution. This representation still involves certain unknown boundary values, thus for this formula to become effective it is necessary to characterize the unknown boundary values in terms of the given boundary conditions. This requires the investigation of certain relations refereed to as the global relations. A general approach for analyzing these relations is illustrated by solving several problems formulated in the interior of a semistrip. In addition, for completeness, similar results are presented for the Poisson equation by employing an integral representation for the Laplace equation which is more general than the one derived in the late 1990s.     

Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method

In this paper, we present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated.     

A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

We consider the approximate solution of axisymmetric biharmonic problems using a boundary-type meshless method, the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation. For such problems, the coefficient matrix of the linear system defining the approximate solution has a block circulant structure. This structure is exploited to formulate a matrix decomposition method employing fast Fourier transforms for the efficient solution of the system. The results of several numerical examples are presented. AMS subject classification 65N38, 65F30, 65T50, 65Y99     

On the convergence of a mixed finite-element method for plate bending problems

Summary In this paper we justify a finite element method for biharmonic boundary value problems. The method is based on a stationary variational principle (the Reissner principle), and was introduced by Hellan, Herrmann and Visser. We prove error estimates and the existence of a finite element solution.     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.