Fourth-order Singular Perturbation Problem Research Articles

The virtual element method with interior penalty for the fourth-order singular perturbation problem

We present the virtual element method with interior penalty to solve a fourth-order singular perturbation problem. In order to estimate the nonconformity error, the degrees of freedom on edges are changed to the moments of functions in the interior penalty scheme. To do this, we design a special H1-type projection that can be uniquely determined by the new degrees of freedom. With the help of the H1-type projection, the local discrete space is defined by imposing some restrictions on the local space of H2-conforming VE, such that it has the local H2 regularity. Then for the numerical method, we derive the error estimates under the condition of enough smooth solution and the uniform error estimates for cases with boundary layers. Finally, we display some numerical examples. From that, we see that the interior penalty method gives a better performance on the convergence than the theoretical predictions.

An asymptotic-preserving finite element method for a forth order singular perturbation problem with boundary layers

We propose an asymptotic-preserving finite element method for a fourth order singular perturbation problem, which completely preserves the asymptotic transition of the underlying partial differential equation. As a representative, we analyze the NZT element and a linear convergence rate is proved for the solution with sharp boundary layer. Numerical examples in two and three dimensions are consistent with the theoretical prediction.

The nonconforming virtual element method for fourth-order singular perturbation problem

We present the nonconforming virtual element method for the fourth-order singular perturbation problem. The virtual element proposed in this paper is a variant of the C0-continuous nonconforming virtual element presented in our previous work and allows to compute two different projection operators that are used for the construction of the discrete scheme. We show the optimal convergence in the energy norm for the nonconforming virtual element method. Further, the lowest order nonconforming method is proved to be uniformly convergent with respect to the perturbation parameter. Finally, we verify the convergence for the nonconforming virtual element method by some numerical tests.

Chebyshev Differential Quadrature for Numerical Solutions of Third- and Fourth-Order Singular Perturbation Problems

In this paper, linear and nonlinear singularly perturbed problems are studied by a numerical approach based on polynomial differential quadrature. The weighting coefficient matrix is acquired using Chebyshev polynomials. Different classes of perturbation problems are considered as test problems to show the accuracy of method. Then, the quadrature results are compared with analytical solutions of well-known existing solutions.

A Robust Double Set Parameter Nonconforming Rectangular Element for Fourth Order Elliptic Singular Perturbation Problems

In this paper, with double set parameter method and bubble function technique, a robust C0 nonconforming rectangular element is proposed for solving fourth order singular perturbation problems. The properties of the new element is studied and the convergent result in the energy norm uniformly with respect to the perturbation parameter is presented. Numerical results are carried out to demonstrate the validity of the theoretical analysis.

High order finite volume methods for singular perturbation problems

In this paper we establish a high order finite volume method for the fourth order singular perturbation problems. In conjunction with the optimal meshes, the numerical solutions resulting from the method have optimal convergence order. Numerical experiments are presented to verify our theoretical estimates.

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ϵ multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.

A robust nonconforming $H^2$-element

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming H 2 -element which is H 1 -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.

Locking in finite-element approximations to long thin extensible beams

Finite-element approximations to a linearized model of the extensible beam problem are examined. For long thin beams, the model may be viewed as a fourth-order singular perturbation problem. If the mesh size in the finite-element method is not taken to be fine enough, the numerical solution may exhibit locking; we explore the cause of this phenomenon and a possible correction.