Problems For Second-order Elliptic Equations Research Articles

Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

The Meyer’s estimate of solutions to Zaremba problem for second-order elliptic equations in divergent form

The Meyer’s estimate of solutions to Zaremba problem for second-order elliptic equations in divergent form

On some asymptotic representations of solutions to elliptic equations and their applications

The question on asymptotic representations of Green's function for classical boundary value problems for second-order elliptic equations depending on a complex parameter is considered. The equations are defined either in a domain with a compact boundary or in the whole space or the half-space. The main asymptotic term is written out. The results can be applied for the study of some inverse problems of recovering point sources in heat and mass transfer problems.

On Solving Second-Order Linear Elliptic Equations

A method is presented for solving interior boundary-value problems for second-order elliptic equations by transition to ray variables. The domain is divided into cells within which the coefficients and sources have the smoothness and continuity properties necessary for the existence of a regular classical solution in the cell. The finite discontinuities of the coefficients (if any) are located on the cell boundaries. The regular solution in the cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on the rays arriving at the given point from the cell boundaries. Next, a finite analytic scheme for the numerical solution of the boundary value problem in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme exhibits no hard dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.

An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients

We develop a numerical approximation involving boundary integral techniques for the solution of the Dirichlet problem for second-order elliptic equations with variable coefficients. Using the concept of a parametrix, the problem is reduced to a boundary-domain integral equation to be solved for two unknown densities. Via a change of variables based on shrinkage of the boundary curve of the solution domain a parameterised system of boundary-domain integrals is obtained. It is shown how to write the singularities in this system in an explicit form such that boundary integral techniques can be applied for analysis and discretisation. An effective discretisation involving the Nystrom method is given, together with numerical experiments showing that the proposed approach can be turned into a practical working method.

An iterative method for the Cauchy problem for second-order elliptic equations

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach.

On the differentiation of the functional in distributed optimization problems with imperfect contact

We investigate issues of numerical solving of optimal control problems for second order elliptic equations with non-self-adjoint operators - convection-diffusion problems. Control processes are described by semi-linear convection-diffusion equation with discontinuous data and solutions (states) subject to the boundary interface conditions of imperfect type (i.e., problems with a jump of the coefficients and the solution on the interface; the jump of the solution is proportional to the normal component of the flux). Controls are involved in the coefficients of diffusion and convective transfer. We prove differentiability and Lipshitz continuity of the cost functional, depending on a state of the system and a control. The calculation of the gradients uses the numerical solutions of direct problems for the state and adjoint problems.

On the Robin Problem for Second-Order Elliptic Equations in Cylindrical Domains

В полубесконечном цилиндре рассматривается поведение обобщенных решений дивергентных эллиптических уравнений второго порядка, удовлетворяющих на боковой поверхности цилиндра третьему краевому условию. Библиография: 10 названий.

THE CORRECTNESS OF THE DIRICHLET PROBLEM IN A CYLINDRICAL DOMAIN FOR DEGENERATE MULTIDIMENSIONAL ELLIPTIC-PARABOLIC EQUATIONS

The correctness of boundary value problems on the plane for elliptic equations by the method of the theory of analytic functions of a complex variable has been well studied. When investigating similar questions, when the number of independent variables is greater than two, problems of a fundamental nature arise. A very attractive and convenient method of singular integral equations loses their validity due to the absence of any full theory of multidimensional singular integral equations. Boundary value problems for second-order elliptic equations in domains with edges have been studied in detail. In the authors papers explicit forms of classical solutions of Dirichlet problems in cylindrical domains for multidimensional elliptic equations are found. In this paper we use the method proposed in the authors works, we show the unique solvability and obtain an explicit form of classical solution of the Dirichlet problem in a cylindrical domain for degenerate multidimensional elliptico-parabolic equations.

Existence of solutions to a nonvariational elliptic boundary value problem with parameter and discontinuous nonlinearity

We consider the question of the existence of the Dirichlet problem for second-order elliptic equations with spectral parameter and a nonlinearity discontinuous with respect to the phase variable. Here it is not assumed that the differential operator is formally selfadjoint. Using the method of upper and lower solutions, we establish results on the existence of nontrivial (positive and negative) solutions under positive values of the spectral parameter for the problems under study.

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n−1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ⊂ ℝ n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W 2 1 (Q) under the additional requirement that the boundary function belongs to the space W 2 1/2 (∂Q).

Coupling at a Distance HDG and BEM

We introduce a new technique for numerically solving exterior Dirichlet boundary-value problems for second-order elliptic equations. It consists of coupling a hybridizable discontinuous Galerkin (HDG) method used for solving the so-called interior problem on a bounded region containing the support of the source term, with a boundary element method (BEM) for solving the problem exterior to that region. The novelty is that the BEM is defined on a suitably chosen, smooth artificial boundary whereas the HDG method is defined on a polyhedral subdomain. Because of the choice of the artificial boundary, we can take advantage of the spectral convergence of the BEM solution and of the simplicity of the corresponding equations. Because the HDG method is defined on a polyhedral subdomain, there is no need to try to fit the mesh to the artificial boundary. Instead, the HDG is coupled at a distance with the BEM by using simple Dirichlet-to-Neumann operators defined in the region between the artificial boundary and the polyhedral subdomain. Numerical experiments displaying the performance of the new technique are presented. Optimal orders of convergence are obtained even though the distance between the artificial boundary and the polyhedral domain is of order $h$.

Solving Dirichlet Boundary-value Problems on Curved Domains by Extensions from Subdomains

We present a technique for numerically solving Dirichlet boundary-value problems for second-order elliptic equations on domains $\Omega$ with curved boundaries. This is achieved by using suitably defined extensions from polyhedral subdomains $\textsf{D}_h$; the problem of dealing with curved boundaries is thus reduced to the evaluations of simple line integrals. The technique is independent of the representation of the boundary, of the space dimension, and allows for the use of only polyhedral elements. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that the convergence properties of the resulting method are the same as those for the case in which $\Omega=\textsf{D}_h$ whenever the distance of $\textsf{D}_h$ to $\partial\Omega$ is of order $h$. In particular, we find that when using polynomial approximations of degree $k$ on the polyhedral subdomains $\textsf{D}_h$, both the scalar and vector unknowns converge with the optimal order of $k+1$ in the whole domain $\Omega$ for $k\ge0$. Moreover, we also find that for $k\ge1$, both a postprocessing of the scalar unknown in the polyhedral subdomain $\textsf{D}_h$ as well as its extension from it converge with order $k+2$. The extension also converges with order $2$ for $k=0$.

A new process for constructing coefficients of elliptic complex equations in multiply connected domains

This article concerns the inverse problem for linear elliptic systems of first-order equations with Riemann–Hilbert-type map in multiply connected domains. First the formulation and the complex form of the problem for the systems are given, and then the coefficients of the elliptic complex equations for the above problem are constructed by a complex analytic method, where the advantage of the methods in other papers is absorbed, and the used method in this article is more simple and the obtained result is more general. As an application of the above results, we can derive the corresponding results of the inverse problem for second-order elliptic equations from Dirichlet to Neumann map in multiply connected domains.

Solvability of the Dirichlet problem for a general second-order elliptic equation

The paper is concerned with studying the solvability of the Dirichlet problem for the second-order elliptic equation in a bounded domain , , with -smooth boundary and boundary condition .Conditions for the existence of an -dimensionally continuous solution are obtained, the resulting solvability condition is shown to be similar in form to the solvability condition in the conventional generalized setting (in ). In particular, the problem is shown to have an -dimensionally continuous solution for all and all and from the appropriate function spaces, provided that the homogeneous problem (with zero boundary conditions and zero right-hand side) has no nonzero solutions in .Bibliography: 14 titles.

$H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains

In this paper we consider estimates of the Raleigh quotient and in general of the $H^{1,p}$-eigenvalue in quasicylindrical domains. Then we apply the results to obtain, by variational methods, existence and uniqueness of weak solutions of the Dirichlet problem for second-order elliptic equations in divergent form. For such solutions global boundedness estimates have been also established.

Numercial solution of boundary-valtue problems for second-order elliptic equations with mixed derivatives by effective iteration methods

The paper considers the numerical solution of boundary-value problems for second-order elliptic equations with mixed derivatives occurring, in particular, in the mathematical simulation of physical processes in the anisotropic medium. The spectral properties of matrices obtained from a center-difference approximation of the problem are investigated. For an approximated solution of the system of linear algebraic equations, a new two-step skew-symmetric iteration method is used.

Domain Decomposition Methods With Overlapping Subdomains For The Time-Dependent Problems Of Mathematical Physics

AbstractAt the present time, the domain decomposition methods are considered as the most promising ones for parallel computer systems. Nowadays success is attained mainly in solving approximately the classical boundary value problems for second-order elliptic equations. As for the time-dependent problems of mathematical physics, there are, in common use, approaches based on ordinary implicit schemes and implemented via iterative methods of the domain decomposition. An alternative technique is based on the non-iterative schemes (region-additive schemes). On the basis of the general theory of additive schemes a wide class of difference schemes (alternative directions, locally one-dimensional, factorized schemes, summarized approximation schemes, vec-tor additive schemes, etc.) as applied to the domain decomposition technique for time-dependent problems with synchronous and asynchronous implementations has been investigated. For nonstationary problems with self-adjoint operators, we have considered three dif-ferent types of decomposition operators corresponding to the Dirichlet and Neumann conditions on the subdomain boundaries. General stability conditions have been obtained for the region-additive schemes. We focused on the accuracy of domain decom-position schemes. In particular, the dependence of the convergence rate on the width of subdomain overlapping has been investigated as the primary property. In the present paper, new classes of domain decomposition schemes for nonstationary problems, based on the subdomain overlaping and minimal data exchange in solving problems in subdomains, have been constructed.

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂ Ω i , where the Ω i are disjoint subregions of the original region Ω , a condition number estimate C ( 1 + max i log ( H i / h i ) ) 2 is established with C independent of h i , H i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.

Bounds for elliptic operators in weighted spaces

Some estimates for solutions of the Dirichlet problem for second-order elliptic equations are obtained in this paper. Here the leading coefficients are locally VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.