Elliptic Singular Perturbation Problems Research Articles

Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

Modified Morley method by free formula for fourth order elliptic singular perturbation problems

A new modified Morley method is proposed to solve fourth order elliptic singular perturbation problems of two or three dimension. The method has the fewest number of DOFs on each element. Different from the usual finite element methods, we slightly modify the bilinear term corresponding to the second order differential operator and keep that corresponding to the fourth order differential operator and the right-hand side term unchanged. The modification is based on the idea of free formula introduced by Bergan and Nygård in 1984. Theory and numerical results show that the solution is uniformly convergent with respect to the perturbed parameter ε . From the view of computation, our method is efficient since the fewest number of DOFs are employed and the local stiffness matrix takes less computation time than the usual finite element method due to the special construction of free formula.

A Morley–Wang–Xu Element Method for a Fourth Order Elliptic Singular Perturbation Problem

A Morley–Wang–Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche’s technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming \(P_1\)–\(P_0\) element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.

Robust nonconforming polynomial finite elements over quadrilaterals

This work proposes two nonconforming polynomial finite elements over general convex quadrilaterals. The first one is designed for fourth order elliptic singular perturbation problems, and the other works for Brinkman problems, approximating the velocity with piecewise constant pressure. We show the robustness of these methods, namely, the discrete solution converges uniformly in the given parameters of the corresponding model problem. Numerical examples are also provided.

Multi-peak positive solutions to a class of Kirchhoff equations

In the present paper, we consider the nonlocal Kirchhoff problem$$-\left(\epsilon^2a+\epsilon b\int_{{\open R}^{3}}\vert \nabla u \vert^{2}\right)\Delta u+V(x)u=u^{p}, \quad u \gt 0 \quad {\rm in} {\open R}^{3},$$ where a, b>0, 1<p<5 are constants, ϵ>0 is a parameter. Under some mild assumptions on the function V, we obtain multi-peak solutions for ϵ sufficiently small by Lyapunov–Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{{\open R}^{3}} \vert \nabla u \vert^{2}\right)\Delta u$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of the elliptic singular perturbation problems.

AC0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem

In this paper, aC0nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateralQ, the shape function space is the union ofS21(Q*) and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.

Partition of unity isogeometric analysis of two dimensional elliptic singular perturbation problems

The design basis functions on the reference domain in IGA are diversified and enhanced by extra enrichment functions and various local refinements with the use of partition of unity (PU) function with flat-top. These reconditioned and modified basis functions are pushed forward to the physical domain by the original design mapping for analysis. With this method, the corresponding stiffness matrix has a small bandwidth and local refinement is simple. Moreover, we construct the PU functions in the reference domain and then move them to a physical domain through a geometric mapping to be used for the generation of global basis functions on a physical domain. Therefore, we also have several advantages in calculating stiffness matrices and load vectors. Here we apply this method to various boundary layer problems.

Uniformly convergent C0‐nonconforming triangular prism element for fourth‐order elliptic singular perturbation problem

In this article, we introduce a C 0‐nonconforming triangular prism element for the fourth‐order elliptic singular perturbation problem in three dimensions by using the bubble functions. The element is proved to be convergent in the energy norm uniformly with respect to the perturbation parameter. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1785–1796, 2014

Uniformly Convergent Nonconforming Element for 3-D Fourth Order Elliptic Singular Perturbation Problem

In this paper, using a bubble function, we construct a cuboid element to solve the fourth order elliptic singular perturbation problem in three dimensions. We prove that the nonconforming C 0 -cuboid element converges in the energy norm uniformly with respect to the perturbation parameter. Mathematics subject classification: 65N12, 65N30.

Convergence Analysis of Incomplete Biquadratic Rectangular Element for Fourth-Order Singular Perturbation Problem on Anisotropic Meshes

The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.

On sensitive elliptic singular perturbation problems and logarithmic oscillations: the case of shells with edges

This work deals with singular perturbation problems depending on small positive parameter ϵ. The limit problem as ϵ → 0 has no solution within the classical theory of PDEs, which uses distribution theory. A very particular and less‐known phenomenon appears: large oscillations. These problems exhibit some kind of instability; very small and smooth variations of the data imply large singular perturbations of the solution. That kind of problems appears in elasticity for highly compressible two‐dimensional bodies and thin shells with elliptic middle surface with a part of the boundary free. Here, we consider certain properties of that oscillations and extend the theory to shells with edges. Copyright © 2012 John Wiley & Sons, Ltd.

Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

AbstractIn this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non‐ C0 rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C0 extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

An equation decomposition method for the numerical solution of a fourth‐order elliptic singular perturbation problem

AbstractIn this article, we propose a tailored finite point method (TFPM) for the numerical solution of a type of fourth‐order singular perturbation problem in two dimensions based on the equation decomposition technique. Our finite point method has been tailored based on the local exponential basis functions. Furthermore, our TFPM satisfies the discrete maximum principle automatically. Our numerical examples show that our method has second order convergence rate in energy norm as $\varepsilon\to0$. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Normally elliptic singular perturbations and persistence of homoclinic orbits

We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow variables. Assuming a steady state persists, we construct the stable, unstable, center-stable, center-unstable, and center manifolds of the steady state of a size of order O(1) and give their leading order approximations. Finally, using these tools, we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems.

A $\mathcal{C}^0$ Interior Penalty Method for a Fourth Order Elliptic Singular Perturbation Problem

In this paper, we develop a $\mathcal{C}^0$ interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains. Using some a posteriori error analysis techniques, we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions. In addition, we analyze the convergence of the numerical solution to the unperturbed second order problem. Finally, we perform some numerical experiments that back up the theoretical results.

A new robust C0-type nonconforming triangular element for singular perturbation problems

In this paper, a new robust C 0 triangular element is proposed for the fourth order elliptic singular perturbation problem with double set parameter method and bubble function technique, and a general convergence theorem for C 0 nonconforming elements is presented. The convergence of the new element is proved in the energy norm uniformly with respect to the perturbation parameter. Numerical experiments are also carried out to demonstrate the efficiency of the new element.

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.

A two phase elliptic singular perturbation problem with a forcing term

We study the following two phase elliptic singular perturbation problem: Δ u ε = β ε ( u ε ) + f ε , in Ω ⊂ R N , where ε > 0 , β ε ( s ) = 1 ε β ( s ε ) , with β a Lipschitz function satisfying β > 0 in ( 0 , 1 ) , β ≡ 0 outside ( 0 , 1 ) and ∫ β ( s ) d s = M . The functions u ε and f ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit ( ε → 0 ) and we show that limit functions are solutions to the two phase free boundary problem: Δ u = f χ { u ≢ 0 } in Ω ∖ ∂ { u > 0 } , | ∇ u + | 2 − | ∇ u − | 2 = 2 M on Ω ∩ ∂ { u > 0 } , where f = lim f ε , in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f ε ≡ 0 . The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.

First Order Approximation of an Elliptic 3D Singular Perturbation Problem

A three‐dimensional elliptic singular perturbation problem with discontinuous boundary values is considered. The solution of the problem is written in terms of a double integral. A saddle point analysis is used to obtain a first approximation, which is expressed in terms of a function that can be viewed as a generalization of the complementary error function.