Order Elliptic Differential Equations Research Articles

The Neumann problem for higher order elliptic equations

We solve the Neumann problem in the half space for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness space where Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in Lp or for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.

Comparing variational methods for the hinged Kirchhoff plate with corners

AbstractThe hinged Kirchhoff plate model contains a fourth order elliptic differential equation complemented with a zeroeth and a second order boundary condition. On domains with boundaries having corners the strong setting is not well‐defined. We here allow boundaries consisting of piecewise C2, 1‐curves connecting at corners. For such domains different variational settings will be discussed and compared. As was observed in the so‐called Saponzhyan–Babushka paradox, domains with reentrant corners need special care. In that case, a variational setting that corresponds to a second order system needs an augmented solution space in order to find a solution in the appropriate Sobolev‐type space.

Combination of the meshless finite difference approach with the Monte Carlo random walk technique for solution of elliptic problems

This paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations. It is based upon the well-known Monte Carlo (MC) method with a random walk (RW) technique, carried out on the grid of points. This method allows for accurate estimation of the solution of the differential equation at selected point(s) of the domain and/or its boundary. It extends the standard formulation of the Monte Carlo–random walk (MC–RW) approach by means of its appropriate combination with the meshless version of the finite difference method. In this manner, the proposed approach may deal with elliptic equations in more general non-homogeneous form as well as boundary conditions of both essential and natural types. Moreover, arbitrarily irregular clouds of nodes may be used, with no a-priori imposed nodes structure. Therefore, the meshless MC/RW approach may be applied to the significantly wider class of problems with more complex geometry.This concept was examined on variety of 2D boundary value problems. Selected numerical results are presented and discussed. A simple Matlab code is included as well.

Hölder Continuity of Solutions for Quasilinear Elliptic Equations with Gradient Terms and Measures

We consider quasi-linear second order elliptic differential equations with gradient terms and study Holder continuity of solutions of the equation. Also, as an application we investigate removable sets for Holder continuous solutions of a certain equation.

A Monotone Iterative Technique for Nonlinear Fourth Order Elliptic Equations with Nonlocal Boundary Conditions

This paper proposes an accelerated iterative procedure for a nonlinear fourth order elliptic equation with nonlocal boundary conditions. First, an existence and uniqueness theorem is proved for the fourth order elliptic equation via the accelerated iterative procedure. To solve this problem numerically, a finite difference based numerical scheme is also developed in view of the main theorem. Theoretically, the monotone property as well as the convergence analysis are proved for both the continuous and discretized cases. The main result also supplements several algorithms for computing the solution of the fourth order elliptic integro-partial differential equation. The proposed scheme not only accelerates the scheme in the literature but also provides a greater flexibility in choosing the initial guess. The efficacy of the proposed scheme is demonstrated through a comparative numerical study with the recent literature. The numerical simulation confirms the theoretical claims too.

High Order Finite Difference Discretization for Composite Grid Hierarchy and Its Applications

AbstractWe introduce efficient approaches to construct high order finite difference discretizations for solving partial differential equations, based on a composite grid hierarchy. We introduce a modification of the traditional point clustering algorithm, obtained by adding restrictive parameters that control the minimal patch length and the size of the buffer zone. As a result, a reduction in the number of interfacial cells is observed. Based on a reasonable geometric grid setting, we discuss a general approach for the construction of stencils in a composite grid environment. The straightforward approach leads to an ill-posed problem. In our approach we regularize this problem, and transform it into solving a symmetric system of linear of equations. Finally, a stencil repository has been designed to further reduce computational overhead. The effectiveness of the discretizations is illustrated by numerical experiments on second order elliptic differential equations.

Removable sets for Hölder continuous solutions of quasilinear elliptic equations with lower order terms

We consider quasi-linear second order elliptic differential equations with lower order terms and study removable sets for Holder continuous solutions of the equation.

An efficient compact difference scheme for solving the streamfunction formulation of the incompressible Navier–Stokes equations

Recently, a new paradigm for solving the steady Navier–Stokes equations using a streamfunction–velocity formulation was proposed by Gupta and Kalita [M.M. Gupta, J.C. Kalita, A new paradigm for solving Navier–Stokes equations: streamfunction–velocity formulation, J. Comput. Phys. 207 (2005) 52–68], which avoids difficulties inherent in the conventional streamfunction–vorticity and primitive variable formulations. It is discovered that this formulation can reached second-order accurate and obtained accuracy solutions with little additional cost for a couple of fluid flow problems. In this paper, an efficient compact finite difference approximation, named as five point constant coefficient second-order compact (5PCC-SOC) scheme, is proposed for the streamfunction formulation of the steady incompressible Navier–Stokes equations, in which the grid values of the streamfunction and the values of its first derivatives (velocities) are carried as the unknown variables. The derivation approach is simple and can be easily used to derive compact difference schemes for other similar high order elliptic differential equations. Numerical examples, including the lid driven cavity flow problem and a problem of flow in a rectangular cavity with the hight–width ratio of 2, are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The results obtained are compared with ones by different available numerical methods in the literature. The present scheme not only shows second-order accurate, but also proves more effective than the existing second-order compact scheme of the streamfunction formulation in the aspect of computational cost.

Schwarz problem for higher-order complex elliptic partial differential equations

In this paper, we investigate the solvability of a higher order elliptic differential equation with the Schwarz conditions on the unit disc of the complex plane.

Second Order Elliptic Equations in $\mathbb{R}^{d} $ with Piecewise Continuous Coefficients

The existence and uniqueness of solutions of second order elliptic differential equations in \(\mathbb{R}^{d} \) are proved. The coefficients of second order terms are allowed to have discontinuity at finitely many parallel hyper-planes in \(\mathbb{R}^{d} \) and the first derivatives of solutions can have jumps at the hyper-planes.

Some new oscillation theorems for second order nonlinear elliptic equations with damping

In this paper, some new oscillation criteria are obtained for second order elliptic differential equations with damping of the form ∇ ⋅ ( A ( x ) ∇ y ) + B T ( x ) ∇ y + q ( x ) f ( y ) = 0 , x ∈ Ω , where Ω is an exterior domain in R N . These criteria are different from most known ones in the sense that they are based on a class of new functions of the form Φ ( r , s , l ) defined in the sequel.

Integral operator and oscillation of second order elliptic equations

By using integral operator, some oscillation criteria for second order elliptic differential equation$$ \sum^d _{i,j=1} D_i[A_{ij}(x)D_jy]+ q(x)f(y)=0, \;x \in \Omega\qquad \eqno{(E)} $$are established. The results obtained here can be regarded as the extension of the well-known Kamenev theorem to Eq.$(E)$.

Some new oscillation criteria for second order elliptic equations with damping

Some new oscillation criteria are obtained for second order elliptic differential equations with damping $$\sum_{i,j=1}^nD_i[A_{ij}(x)D_jy]+\sum_{i=1}^nb_i(x)D_iy+q(x)f(y)=0,\ \quad x\in{\mit\Omega},$$ where ${\mit\Omega} $ is an exterior domain in $\math

Elastic potentials at corners in Sobolev spaces with asymptotics

AbstractThis paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.

Schauder estimates for solutions to boundary value problems for second order elliptic systems in polyhedral domains

Boundary value problems for second order elliptic differential equations and systems in a polyhedral domain are considered. The authors prove Schauder estimates and obtain regularity assertions for the solutions.

Maximum estimates for oblique derivative problems with right hand side in L p , p < n

If u is a solution of the second order elliptic differential equation Lu=f in some Lipschitz domain in ℝn, we estimate the maximum of u in terms of some oblique derivative prescribed on the boundary of the domain and in terms of the Lp norm of f with p<n.

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.

Problems for elliptic equations with operator-boundary conditions

We consider, in a Hilbert space H, a problem on [0,1] for a second order elliptic operator-differential equation with operator-boundary conditions. We also consider second order elliptic differential equations with operatorboundary conditions in cylindrical domains in the case when operator-boundary conditions contain integral terms over the whole domain. In this case, the proof of the density of the domain of definition of operators in a space is difficult. When boundary conditions are local, this fact is a simple corollary of the density ofC 0 ∞ (Ω) inL p (Ω).

A Posteriori Error Estimates for Finite Volume Element Approximations of Convection–Diffusion–Reaction Equations

We present the results of a study on a posteriori error control strategies for finite volume element approximations of second order elliptic differential equations. Finite volume methods ensure local mass conservation and, combined with some up-wind strategies, give monotone solutions. We adapt the local refinement techniques known from the finite element method to the finite volume discretizations of various boundary value problems for steady-state convection–diffusion–reaction equations. In this paper we derive and study a residual type error estimator and illustrate its practical performance on a series of computational tests in 2 and 3 dimensions. Our tests show that the discussed locally conservative approximation methods with a posteriori error control can be used successfully in numerical simulation of fluid flow and transport in porous media.

Existence and uniqueness of the solution, separation for certain second order elliptic differential equation

In this paper, we study conditions under which Schrodinger type operators with a matrix potential is separated and Schrodinger equation has a unique solution in the weighted space L2,k(Rn)l, where l is any natural number and k ε C1(Rn) is a positive function