Elliptic Boundary Value Problems Research Articles

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

An adaptively enriched coarse space for Schwarz preconditioners forP1 discontinuous Galerkin multiscale finite element problems

AbstractIn this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.

An efficient extrapolation full multigrid method for elliptic problems in two and three dimensions

An extrapolation full multigrid (EXFMG) method is proposed for solving the large linear systems arising from linear finite element discretization of two- (2D) and three-dimensional (3D) elliptic boundary value problems. A good initial guess on the next finer grid is constructed through combining Richardson extrapolation and quadratic FE interpolation for the numerical solutions on two-level of grids (current and previous grids). And the FE linear system on the next finer grid is solved to obtain the EXFMG solution with discretization-level accuracy by performing one or two multigrid cycles with Gauss–Seidel smoother. Moreover, Richardson extrapolation is further used to achieve higher-order accuracy. The numerical results of five typical elliptic problems including one 2D problem and four 3D problems are conducted to show the effectiveness and efficiency of the EXFMG algorithm.

Fractional-power approach for the study of elliptic second-order boundary-value problems with variable-operator coefficients in an unbounded domain

In this article, we give new results on the study of elliptic complete abstract second order differential equations with variable operator coefficients under Dirichlet boundary conditions, and set in \(\mathbb{R}_+\). In the framework of Holderian spaces and under some compatibility conditions, we prove the main results on the existence, uniqueness and maximal regularity of the classical solution of this kind of problems which have not been studied in variable coefficients case. We use semigroups theory, fractional powers of linear operators, Dunford's functional calculus and interpolation theory. In this work, we consider some differentiability assumptions on the resolvents of square roots of linear operators.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/89/abstr.html

A unified analysis of a class of quadratic finite volume element schemes on triangular meshes

This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H1 error estimate is proved by the standard techniques.

Iterative solution methods for elliptic boundary value problems

AbstractFor elliptic boundary value problems (the diffusion equation and elasticity theory ones) with highly varying coefficients, there are proposed iterative methods with the number of iterations independent of the coefficient jumps. In the differential case these methods take solving the boundary value problem for the Poisson equation at each step of iterations while in the finite difference (finite element) approximation it is possible to use another operator as a preconditioner.

Convergence of a generalized penalty method for variational–hemivariational inequalities

The aim the paper is to study a large class of variational-hemivariational inequalities involving constraints in a Banach space. First, we establish a general existence theorem for this class. Second, we introduce a sequence of penalized problems without constraints. Under the suitable assumptions, we prove that the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems, w--lim supn→∞Sn, is nonempty and is contained in the set of solutions to original inequality problem. Also, we prove the identity, w--lim supn→∞Sn=s--lim supn→∞Sn, when operator A satisfies (S)+-property. Finally, we illustrate the applicability of the theoretical results and we explore two complicated partial differential systems of elliptic type, which are an elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints, respectively.

Semilinear Fractional Elliptic Problems with Mixed Dirichlet-Neumann Boundary Conditions

Abstract We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.

Condition numbers of finite element methods on a class of anisotropic meshes

We study the behavior of the finite element condition numbers on a class of anisotropic meshes. These newly-developed mesh algorithms can produce numerical approximations with optimal convergence to isotropic and anisotropic singular solutions of elliptic boundary value problems in two- and three-dimensions. Despite the simplicity and fewer geometric constraints in implementation, these meshes can be highly anisotropic and do not maintain the maximum angle condition. We formulate a unified refinement principle and establish sharp estimates on the growth rate of the condition numbers of the stiffness matrix from these meshes. These results are important for effective applications of these meshes and for the design of fast numerical solvers. Numerical tests validate the theoretical analysis.

The Continuous Classical Optimal Control Problems for Triple Nonlinear Elliptic Boundary Value Problem

In this research, our aim is to study the optimal control problem (OCP) for triple nonlinear elliptic boundary value problem (TNLEBVP). The Mint-Browder theorem is used to prove the existence and uniqueness theorem of the solution of the state vector for fixed control vector. The existence theorem for the triple continuous classical optimal control vector (TCCOCV) related to the TNLEBVP is also proved. After studying the existence of a unique solution for the triple adjoint equations (TAEqs) related to the triple of the state equations, we derive The Fréchet derivative (FD) of the cost function using Hamiltonian function. Then the theorems of necessity conditions and the sufficient condition for optimality of the constraints problem are proved

An iterative technique for solving a class of local and nonlocal elliptic boundary value problems

An optimal iterative method is proposed for a reliable solution of a class of Bratu-type, Troesch’s and nonlocal elliptic boundary value problems (BVPs). Due to the presence of parameter $$\delta $$ as well as strong nonlinearity, these problems pose difficulties in obtaining their solutions. With the help of Green’s function theory, we first transform the BVP into an equivalent integral equation, followed by applying the optimal homotopy analysis method to get the approximate solution of high accuracy level. Several examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. The convergence and error analysis of the proposed method is discussed.

New Positive Solutions of Nonlinear Elliptic PDEs

We are concerned with positive solutions of two types of nonlinear elliptic boundary value problems (BVPs). We present conditions for existence, uniqueness and multiple positive solutions of a first type of elliptic BVPs. For a second type of elliptic BVPs, we obtain conditions for existence and uniqueness of positive global solutions. We employ mathematical tools including strictly upper (SU) and strictly lower (SL) solutions, iterative sequence method and Amann theorem. We present our research findings in new original theorems. Finally, we summarize and indicate areas of future study and possible applications of the research work.

The optimal convergence of finite element methods for the three-dimensional second-order elliptic boundary value problem with singularity

Let x0∈Ω. Assume that Gx0(x) satisfies the elliptic boundary value problem LGx0(x)≡∂∂xj(aij(x)∂Gx0(x)∂xi)=δ(x−x0),inΩ,Gx0(x)=0on∂Ω, where Ω⊂ℜ3 is a convex polyhedral domain. In this article, we propose specially graded meshes Th. Furthermore, we obtain an L2(Ω)-error estimate of order hk+1|ln⁡h|32 for approximations with a Pk finite element method.

Principal eigenvalues of elliptic BVPs with glued Dirichlet-Robin mixed boundary conditions. Large potentials on the boundary conditions

In this paper we perform a complete study about the existence, uniqueness and simplicity of the principal eigenvalue of a class of elliptic eigenvalue problems with glued Dirichlet-Robin boundary conditions on a component of the boundary of the domain and Dirichlet boundary conditions on the other component of the boundary. Moreover, this principal eigenvalue and its normalized principal eigenfunction are approached in R×H1(Ω) by principal eigen-pairs of boundary eigenvalue problems with classical mixed boundary conditions. In some sense, the glued mixed boundary conditions analyzed in this work may be regarded as classical mixed boundary conditions with large potentials in the Robin boundary condition. The principal eigenvalues analyzed in this paper, play a crucial role in analyzing the existence, uniqueness and asymptotic behavior of the positive solutions of certain kinds of semilinear elliptic boundary value problems with nonlinear boundary conditions and spatial heterogeneities. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results in [2], [3] and [4] play a crucial role to obtain some of the results of this work.

On the Existence of Three Nontrivial Solutions of a Resonance Elliptic Boundary Value Problem with a Discontinuous Nonlinearity

We study the homogeneous Dirichlet problem for a second-order elliptic equation with a nonlinearity discontinuous in the state variable in the resonance case. A class of resonance problems that does not overlap with the previously investigated class of strongly resonance problems is singled out. Using the variational method, we establish a theorem on the existence of at least three nontrivial solutions of the problem under study (the zero is its solution). In this case, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on the discontinuities of the nonlinearity only on a set of measure zero. We give an example of a nonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition is obtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest. Applications of the theorem to problems with a parameter are considered. The existence of nontrivial (including semiregular) solutions of the problem with a parameter for an elliptic equation with a discontinuous nonlinearity for all positive values of the parameter is established.

Index of Elliptic Boundary Value Problems Associated with Isometric Group Actions

Given a compact manifold with boundary, endowed with an isometric action of a discrete group of polynomial growth, we state an index theorem for elliptic elements in the algebra of nonlocal operators generated by the Boutet de Monvel algebra of pseudodifferential boundary value problems on the manifold and the shift operators associated with the group action.

Asymptotic Expansions at Nonsymmetric Cuspidal Points

We study the asymptotics of solutions to the Dirichlet problem in a domain $$\mathcal{X} \subset \mathbb{R}^3$$ whose boundary contains a singular point $$O$$ . In a small neighborhood of this point, the domain has the form $$\{ z > \sqrt{x^2 + y^4} \}$$ , i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat’ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.

A novel adaptive finite element method for the ground state solution of Bose-Einstein condensates

Abstract In this study we propose a novel adaptive finite element method for the ground state solution of Bose-Einstein Condensates (BEC). Different from the classical adaptive scheme applied to BEC which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space, our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low-dimensional correction space. Further, the linear elliptic boundary value problem is solved by adaptive multigrid iterations. Since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces, the solving efficiency can be improved evidently. In addition, the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically.

Existence of renomalized solution for nonlinear elliptic boundary value problem without $$\Delta _{2}$$ -condition

In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function $$ \varphi $$ doesn’t satisfy the $$\Delta _{2}$$ condition while the right hand side f belongs to $$W^{-1} E_{\psi }(\Omega )$$ .

On a class of elliptic boundary-value problems with parameter and discontinuous non-linearity

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting , we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray , where . As an application, we consider the Gol’dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.