Equations Of Elliptic Type Research Articles

Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

Novel numerical method based on cubic B-splines for a class of nonlinear generalized telegraph equations in irregular domains

In this work, a cubic B-spline method based on finite difference and meshless approaches for solving 2D generalized telegraph equations in irregular single and multi-connected domains is presented. The three-layer Crank-Nicolson time-stepping scheme is applied for temporal derivatives and systems of second order elliptic partial differential equations (EPDEs) of general type with mixed derivatives and variable coefficients are obtained. The Dirichlet and Robin boundary conditions are considered. The approximate solution on each time layer is sought as series over basis functions which are taken in the form of the tensor products of cubic B-splines. The coefficients of the series are chosen to satisfy the governing EPDEs in the solution domain. The stability and accuracy of the proposed scheme is demonstrated by several examples.

Bifurcation results for a Kirchhoff type problem involving sign-changing weight functions

In this paper we establish some existence results for both compact and unbounded connected components of the positive solutions set of a nonlocal problem of the Kirchhoff type. The methods to show our main results are the local and global bifurcation techniques, a priori bounds for elliptic equation, and the properties of the principal eigenvalues. To obtain the a priori bounds for Kirchhoff type elliptic equation, some new techniques are introduced.

Existence of solutions for a nonhomogeneous elliptic Kircchoff type equation

For a bounded domain Ω in RN, a bounded Carathéodory function g in Ω×R, and a nonnegative and nonzero locally integrable measurable function h in Ω, we give sufficient conditions on the C1-function M:[0,∞)→R for the existence of a solution of the nonlocal elliptic problem{−(1+γM′(∫Ω|∇u|2))Δu=λu+g(x,u)−h(x)|u|p−1u+f(x),x∈Ωu(x)=0,x∈∂Ω for every p>1, γ>0, λ∈R and f∈L2(Ω).

ON THE CAUCHY PROBLEM FOR MATRIX FACTORIZATIONS OF THE HELMHOLTZ EQUATION

In the paper it is considered the regularization of the Cauchy problem for systems of elliptic type equations of the first order with constant coefficients factorisable Helmholtz operator in two-dimensional unbounded domain. Using the results of [20, 21, 22, 23, 24, 25 ], we construct in explicit form Carleman matrix and, based on the regularized solution of the Cauchy problem.

Solutions for fourth-order Kirchhoff type elliptic equations involving concave–convex nonlinearities in [formula omitted

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations Δ2u−M(‖∇u‖22)Δu+V(x)u=f(x,u),x∈RN,where M(t):R→R is the Kirchhoff function, f(x,u)=λk(x,u)+h(x,u), λ≥0, k(x,u) is of sublinear growth and h(x,u) satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ=0. For λ>0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f(x,u) is odd in u, we show that above equations possess infinitely many solutions for all λ≥0. Our theorems generalize some known results in the literatures even for λ=0 and our proof is based on the variational methods.

About one Approach to a Solution of Linear Differential Equations with Variable Coefficients

Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.

Matrix-free multigrid block-preconditioners for higher order discontinuous Galerkin discretisations

Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering, including for example computational fluid dynamics, nuclear reactor simulations and combustion models. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. This includes both distributed- or shared memory parallelisation between processors and vectorisation on individual cores.The computationally most expensive components of the solver are the repeatedapplications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order Discontinuous Galerkin (DG) space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark.

On a nonlinear elliptic problems having large monotonocity with L 1-data in weighted Orlicz-Sobolev spaces

Abstract We prove in weighted Orlicz-Sobolev spaces, the existence of entropy solution for a class of nonlinear elliptic equations of Leray-Lions type, with large monotonicity condition and right hand side f ∈ L 1(Ω).

Finite difference methods for 2D shallow water equations with dispersion

Abstract Basic properties of some finite difference schemes for two-dimensional nonlinear dispersive equations for hydrodynamics of surface waves are considered. It is shown that stability conditions for difference schemes of shallow water equations are qualitatively different in the cases the dispersion is taken into account, or not. The difference in the behavior of phase errors in one- and two-dimensional cases is pointed out. Special attention is paid to the numerical algorithm based on the splitting of the original system of equations into a nonlinear hyperbolic system and a scalar linear equation of elliptic type.

Symmetric results of a Hénon type elliptic equation

In this paper, we study the following weighted elliptic system −Δu+u=λ1|x|αu3+μ|x|βuv2,x∈B,−Δv+v=λ2|x|αv3+μ|x|βu2v,x∈B,u,v>0,x∈B,u=v=0,x∈∂B,where B⊂RN(N=2,3) is the unit ball centered at the origin, λ1,λ2>0, μ>0, β>0, α>0. By virtue of variational approaches and rescaling methods, the system has a nontrivial ground state solution with α>β>0, moreover, by reduction methods, the ground state solution is radial symmetry if β>0 small enough.

SLAC – a semi-Lagrangian artificial compressibility solver for steady-state incompressible flows

PurposeThe purpose of this paper is the development of a new density-based (DB) semi-Lagrangian method to speed up the conventional pressure-based (PB) semi-Lagrangian methods.Design/methodology/approachThe semi-Lagrangian-based solvers are typically PB, i.e. semi-Lagrangian pressure-based (SLPB) solvers, where a Poisson equation is solved for obtaining the pressure field and ensuring a divergence-free flow field. As an elliptic-type equation, the Poisson equation often relies on an iterative solution, so it can create a challenge of parallel computing and a bottleneck of computing speed. This study proposes a new DB semi-Lagrangian method, i.e. the semi-Lagrangian artificial compressibility (SLAC), which replaces the Poisson equation by a hyperbolic continuity equation with an added artificial compressibility (AC) term, so a time-marching solution is possible. Without the Poisson equation, the proposed SLAC solver is faster, particularly for the cases with more computational cells, and better suited for parallel computing.FindingsThe study compares the accuracy and the computing speeds of both SLPB and SLAC solvers for the lid-driven cavity flow and the step-flow problems. It shows that the proposed SLAC solver is able to achieve the same results as the SLPB, whereas with a 3.03 times speed up before using the OpenMP parallelization and a 3.35 times speed up for the large grid number case (512 × 512) after the parallelization. The speed up can be improved further for larger cases because of increasing the condition number of the coefficient matrixes of the Poisson equation.Originality/valueThis paper proposes a method of avoiding solving the Poisson equation, a typical computing bottleneck for semi-Lagrangian-based fluid solvers by converting the conventional PB solver (SLPB) to the DB solver (SLAC) through the addition of the AC term. The method simplifies and facilitates the parallelization process of semi-Lagrangian-based fluid solvers for modern HPC infrastructures, such as OpenMP and GPU computing.

Complete flux scheme for elliptic singularly perturbed differential–difference equations

In this study, we propose a new scheme named as complete flux scheme (CF-scheme) based on the finite volume method for solving singularly perturbed differential–difference equations (SPDDEs) of elliptic type. An alternate integral representation for the flux is obtained which plays an important role in the derivation of CF-scheme. We have established the stability, consistency and quadrature convergence of the proposed scheme. The scheme is successfully implemented on test problems.

On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent$$\left\{\begin{array}{ll}\Delta^2_{p(x)}u-M\left(\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\right)\Delta_{p(x)} u = \lambda f(x,u) \quad \text{ in }\Omega,\\u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right.$$where $\Omega \subset \R^N$, $N \geq 3$, is a smooth bounded domain, $M(t)=a+bt^\kappa$, $a, \kappa>0$, $b \geq 0$, $\lambda$ is a positive parameter, $\Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u)$ is the operator of fourth order called the $p(x)$-biharmonic operator, $\Delta_{p(x)}u = \operatorname{div} \left(|\nabla u|^{p(x)-2}\nabla u\right)$ is the $p(x)$-Laplacian, $p:\overline\Omega \to \R$ is a log-H\"{o}lder continuous function and $f: \overline\Omega\times \R\to \R$ is a continuous function satisfying some certain conditions. Using Ekeland's variational principle combined with variational techniques, an existence result is established in an appropriate function space.

On the implementation of finite deformation gradient-enhanced damage models

We introduce a comprehensive framework for the efficient implementation of finite deformation gradient-regularised damage formulations in existing finite element codes. The numerical implementation is established within a thermo-mechanically fully coupled finite element formulation, where the heat equation solution capabilities are utilised for the damage regularisation. The variationally consistent, gradient-extended and geometrically non-linear damage formulation is based on an overall free energy function, where the standard local free energy contribution is additively extended by two non-local terms. The first additional term basically contains the referential gradient of the non-local damage variable. Secondly, a penalty term is added to couple the local damage variable—the evolution of which is governed by an ordinary differential equation—and the non-local damage field variable that is governed by an additional balance equation of elliptic type.

The Doubly Periodic “Jump” Problem for a Second-Order Linear Elliptic Equation With Constant Coefficients

For a homogeneous differential second-order equation of elliptic type with constant coefficients, we study the boundary-value problem of finding doubly periodic solutions for given jumps of the solutions themselves and their derivatives on a smooth as well as on a non-rectifiable closed contour. We get the necessary and sufficient conditions for the solvability of the problem and obtain explicit formulas for its solution.

A pressure-based algorithm for internal compressible turbulent flows through a geometrical singularity

Compressible turbulent flow through the abrupt enlargement in pipes is studied numerically by means of Advection Upstream Splitting Method (AUSM+-up). In low Mach numbers, a pressure correction equation of elliptic type is derived. This equation is compatible with the nature of governing equations and retrieves hyperbolic characteristic at higher Mach numbers. It is shown that the proposed numerical algorithm is computationally more efficient than the preconditioned density-based methods. The flow parameters such as reattachment length, pressure loss coefficient and wall shear stress are predicted. It is found that the loss coefficient of the compressible flow rises drastically with increasing Reynolds number while it is constant for incompressible flows. Furthermore, the total-pressure ratio drops with increasing Reynolds number and expansion ratio where it approaches an asymptotic curve. In compressible flow, the pressure is minimum and constant after the enlargement section up to the axial position of the recirculation center.

Regularity Bootstrapping for 4th-order Nonlinear Elliptic Equations

Abstract We consider nonlinear 4th-order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are $C^{2,\alpha }$ enjoy interior estimates on all derivatives.

A substructuring preconditioner with vertex-related interface solvers for elliptic-type equations in three dimensions

In this paper, we propose a variant of the substructuring preconditioner for solving three-dimensional elliptic-type equations with strongly discontinuous coefficients. In the proposed preconditioner, we use the simplest coarse solver associated with the finite element space induced by the coarse partition and construct inexact interface solvers based on overlapping domain decomposition with small overlaps. This new preconditioner has an important merit: its construction and efficiency do not depend on the concrete form of the considered elliptic-type equations. We apply the proposed preconditioner to solve the linear elasticity problems and Maxwell’s equations in three dimensions. Numerical results show that the convergence rate of PCG method with the preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficients in the considered equations.

Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques, and set-valued analysis.