The Morley element for fourth order elliptic equations in any dimensions

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general n-dimensional Morley element consists of all quadratic polynomials defined on each n-simplex with degrees of freedom given by the integral average of the normal derivative on each (n-1)-subsimplex and the integral average of the function value on each (n-2)-subsimplex. Explicit expressions of nodal basis functions are also obtained for this element on general n-simplicial grids. Convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation. The work was supported by the National Natural Science Foundation of China (10571006). This work was supported in part by NSF DMS-0209497 and NSF DMS-0215392 and the Changjiang Professorship through Peking University.     

Nonconforming tetrahedral finite elements for fourth order elliptic equations

This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operators in three spatial dimensions. The newly constructed elements include two nonconforming tetrahedral finite elements and one quasi-conforming tetrahedral element. These elements are proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element, while the nonmodified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special grid.

     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations

We prove the global existence and uniqueness of a classical solution to initial boundary value problem for a class of Sobolev type equations under the Dirichlet boundary conditions. This class of evolution equations covers the well-known viscous Cahn-Hilliard equation and the viscous Camassa-Holm equation.     

Heisenberg群上四阶非线性次椭圆方程的可解性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｆｏｕｒｔｈｏｒｄｅｒｓｅｍｉｌｉｎｅａｒｓｕｂｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍΔ2 Ｈｕ +ｃΔＨｕ =ｆ( (ｚ ,ｔ) ,ｕ)　ｉｎＤ ,ｕ｜Ｄ =ΔＨｕ｜Ｄ =0 ,( 1 .1 )ｗｈｅｒｅＤｉｓａｂｏｕｎｄｅｄｏｐｅｎｓｕｂｓｅｔｏｆｔｈｅＨｅｉｓｅｎｂｅｒｇｇｒｏｕｐＨｎａｎｄΔＨｉｓｔｈｅｓｕｂｅｌｌｉｐｔｉｃＬａｐｌａ ｃｉａｎｏｎＨｎ.ＷｅｒｅｃａｌｌｔｈａｔＨｎｉｓｔｈｅＬｉｅｇｒｏｕｐｗｈｏｓｅｕｎｄｅｒｌｙｉｎｇｍａｎｉ…     

Multiple existence of non-radial solutions with group invariance for sublinear elliptic equations

We deal with sublinear elliptic equations in a ball and prove the existence of infinitely many solutions which are not radially symmetric but G invariant. Here G is any closed subgroup of the orthogonal group and is not transitive on the unit sphere.     

On the coupled systems of second and fourth order elliptic equations

In this paper, by the variational method, we discuss the existence and uniqueness of solutions for a coupled system of second and fourth order elliptic equations in the cases where the perturbation is zero, sublinear and superlinear, respectively.     

Maximum principles and bounds for a class of fourth order nonlinear elliptic equations

We deduce maximum principles for a class of fourth order nonlinear elliptic equations by using auxiliary functions containing the square of the second gradient of the solution of such equations. A priori bounds on various quantities of interest are obtained.     

Mountain pass solutions for the coupled systems of second and fourth order elliptic equations

In this paper, by the mountain pass theorem, we give an existence theorem of nontrivial solutions for a coupled system of second and fourth order elliptic equations.     

A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

A fourth algebraic order exponentially-fitted Runge-Kutta method for the numerical solution of the Schrodinger equation

An exponentially-fitted Runge–Kutta method for the numericalintegration of the radial Schrödinger equation is developed.Theoretical and numerical results obtained for the well knownWoods–Saxon potential show the efficiency of the new method.     

Averaging technique for the oscillation of second order damped elliptic equations

By using the averaging technique, we establish some oscillation theorems for the second order damped elliptic differential equation N↑∑↓i,j=1 Di[AIY（x）Djy]＋N↑∑↓i=1 bi（x）Diy＋c（x）f（y）=0 which extend and improve some known results in the literature.     

An efficient algorithm for the parallel solution of high-dimensional differential equations

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further, we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16.     

Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse

The present paper considers the Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse. No restrictions on the multiplicities of the roots of the characteristic polynomial are assumed.     

Positive solution for a class of nonlocal elliptic equations

In this paper, we devote ourselves to investigating the existence of positive solution for a class of nonlocal elliptic equations. Our approach is based on the fixed point index theory.     

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical study using explicit multistep Galerkin finite element method for the MRLW equation

In this article, an explicit multistep Galerkin finite element method for the modified regularized long wave equation is studied. The discretization of this equation in space is by linear finite elements, and the time discretization is based on explicit multistep schemes. Stability analysis and error estimates of our numerical scheme are derived. Numerical experiments indicate the validation of the scheme by L₂– and L_∞– error norms and three invariants of motion.4 © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1875–1889, 2015     

Numerical solution of a class of complex differential equations by the Taylor collocation method in elliptic domains

An approximate method for solving higher‐order linear complex differential equations in elliptic domains is proposed. The approach is based on a Taylor collocation method, which consists of the matrix represantation of expressions in the differential equation and the collocation points defined in an elliptic domain. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010