Schemes For Boundary Value Problems Research Articles

A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy

A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations.

Stability and convergence of difference schemes for boundary value problems for the fractional-order diffusion equation

A family of difference schemes for the fractional-order diffusion equation with variable coefficients is considered. By the method of energetic inequalities, a priori estimates are obtained for solutions of finite-difference problems, which imply the stability and convergence of the difference schemes considered. The validity of the results is confirmed by numerical calculations for test examples.

Convergence of collocation schemes for boundary value problems in nonlinear index 1 DAEs with a singular point

We analyze the convergence behavior of collocation schemes ap- plied to approximate solutions of BVPs in nonlinear index 1 DAEs, which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity in the inherent nonlinear ODE system. In particular, we focus on the case when the inherent ODE system is singular with a singularity of the rst kind and apply polynomial collocation to the original DAE system. We show that for a certain class of well-posed boundary value problems in DAEs having a sufficiently smooth solution, the global error of the collocation scheme converges uniformly with the so-called stage order. Due to the sin- gularity, superconvergence at the mesh points does not hold in general. The theoretical results are supported by numerical experiments.

A finite element method scheme for boundary value problems with noncoordinated degeneration of input data

A finite element method scheme is constructed for boundary value problems with noncoordinated degeneration of input data and singularity of a solution. We look at a rate with which an approximate solution by the proposed finite element method converges toward an exact Rν-generalized solution in the weight set W2,ν*+β2+1/1(Ω, δ), and establish estimates for the finite element approximation.

Two-point boundary value problems, Green's functions, and product integration

When the Green's function for a two-point boundary value problem can be found, the solution for any forcing term reduces to a quadrature. Here we investigate using this as the basis for numerical schemes for boundary value problems and parabolic initial-boundary value problems. Application of Gaussian quadrature is disappointing, only converging slowly due to the discontinuous derivative of the Green's functions; however, rapid convergence is recovered by the use of product integration; that is, the prior accurate evaluation of Green's integral for a set of basis functions and the subsequent evaluation for arbitrary forcing functions by a matrix-vector multiplication. The method requires more operations than finite differencing for the same number of nodes, but converges far more rapidly. For small numbers of nodes, the performance is similar to that for orthogonal collocation; however, at high resolution, collocation accuracy deteriorates due to ill-conditioning whereas the present method is stable.

Integral transformation of elliptic problems within irregular domains

Employs the integral transform method in the hybrid numerical‐analytical solution of fully developed laminar flow within a class of irregularly shaped ducts, with respect to the co‐ordinate system chosen to represent the geometry under consideration. A quite general formulation of a two‐dimensional steady‐state diffusion problem is initially considered, and a formal solution is provided. The original partial differential equation is analytically transformed into an infinite system of ordinary differential equations for the transformed velocity field in the flow direction. On truncation to a sufficiently large finite order, adaptively chosen to meet prescribed accuracy requirements, well‐established numerical schemes for boundary value problems are utilized, readily available in scientific subroutines libraries. Illustrates convergence rates for a few typical duct geometries and critically examines previously reported numerical solutions.

A shooting scheme for boundary-value problems

A systematic shooting scheme is developed to solve a cascade of boundary value problems obtained from a small-parameter expansion of the full partial differential system. The sequence of solutions and the associated solvability conditions can be obtained simultaneously by a method without having to solve an adjoint boundary problem independently.

A finite element method for first order elliptic systems in three dimensions

We consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions. We also discuss div-curl type systems in three independent variables. Both a least squares method and a new subdomain-collocation-least-squares method are considered. The analysis yield optimal asymptotic H 1(Ω) and L 2(Ω) error estimates.

High-order three-point schemes for boundary value problems. II: nonlinear problems

A highly accurate method for solving nonlinear boundary value problems of the convection-diffusion type is presented. It is shown that the rate of convergence of the present method is very rapid, superior to that of the classical central difference methods. Its main feature is that the converged solution can be achieved within a reasonable amount of iterations for any small value of the mesh diffusion coefficient. All these features are supported by comparisons that are presented for several example problems.

Collocation for Singular Perturbation Problems I: First Order Systems with Constant Coefficients

The application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated. Collocation at Gauss, Radau and Lobatto points is considered, for both initial and boundary value problems for first order systems with constant coefficients. Particular attention is paid to symmetric schemes for boundary value problems; these problems may have boundary layers at both interval ends. .br Our analysis shows that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers'' width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, is used in between. Ways to construct appropriate layer meshes are proposed. Of all methods considered, the Lobatto schemes appear to be the most promising class of methods, as they essentially retain their usual superconvergence power for the smooth, reduced solution, whereas Gauss-Legendre schemes do not. .br We also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem. For a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number, which indicates that these systems can be solved safely by Gaussian elimination with scaled partial pivoting.

Convergence of multigrid iterations applied to difference equations

Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel’s iteration as smoothing procedure.

Galerkin Approximations and the Optimization of Difference Schemes for Boundary Value Problems

Corresponding to each finite difference method for approximating the solution of a boundary value problem, there exists an error operator. The norm of this operator is a functional of the data of the problem and is parametrized by the finite difference method. We treat the problem of mini-mizing this norm over the class of difference methods. In addition, an asymptotic formulation of this problem is made. We show that an appropriate Galerkin approximation to the boundary value problem solves the minimization problem. The key feature of the solution procedure is the existence of a mesh-dependent subspace of the Hilbert space in which the boundary value problem is set. If the data of the problem is replaced by an interpolant in this subspace and then if a Galerkin approximation is made in this subspace to the interpolated problem, there results a finite difference scheme which solves the optimization problem. Although the solution to the optimization problem may not be unique, the advantage of our procedure is that examples of optimal difference methods are readily constructed.