The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R ^m , m?>?2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.     

Solution of the Dirichlet Problem for the Laplace equation

For open sets with a piecewise smooth boundary it is shown that a solution of the Dirichlet problem for the Laplace equation can be expressed in the form of the sum of the single layer potential and the double layer potential with the same density, where this density is given by a concrete series.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

Solution of the Robin problem for the Laplace equation

For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.     

Convergence of a numerical scheme of the type of the vortex loop method on a closed surface with an approximation to the surface shape

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises when solving the Neumann boundary value problem for the Laplace equation with the use of the representation of the solution in the form of a double layer potential. We study the case in which an exterior or interior boundary value problem is solved in a domain whose boundary is a smooth closed surface and the integral equation is written out on that surface. For the numerical solution of the integral equation, the surface is approximated by spatial polygons whose vertices lie on the surface. We construct a numerical scheme for solving the integral equation on the basis of such an approximation to the surface with the use of quadrature formulas of the type of the method of discrete singularities with regularization. We prove that the numerical solutions converge to the exact solution of the hypersingular integral equation uniformly on the grid.     

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated.     

Numerical investigation of a model problem for the Poisson equation with inequality constraints in a domain with a cut

A model problem is considered for the Poisson equation in a two-dimensional domain with a cut. The Dirichlet and Neumann conditions are imposed on the exterior boundary of the domain together with the nonnegativity condition for the jump across the edges of the cut. In addition, the absolute value of the gradient inside the domain must be bounded by some constant. The boundary value problem turns into a variational problem, and the unknown function must yield the minimum of the energy functional on some convex set. After discretization of the problem by the finite element method, an Uzawa-type algorithm is used to find a solution. Some examples are included of solving the discrete problem.     

On the numerical method for solving a hypersingular integral equation with the computation of the solution gradient

We consider a linear integral equation, which arises when solving the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a double layer potential, with a hypersingular integral treated in the sense of Hadamard finite value. We consider the case in which the exterior or interior problem is solved in a domain whose boundary is a closed smooth surface and the integral equation is written over that surface. A numerical scheme for solving the integral equation is constructed with the use of quadrature formulas of the type of the method of discrete singularities with a regularization for the use of an irregular grid. We prove the convergence, uniform over the grid points, of the numerical solutions to the exact solution of the hypersingular equation and, in addition, the uniform convergence of the values of the approximate finite-difference derivative operator on the numerical solution to the values on the projection of the exact solution onto the subspace of grid functions with nodes at the collocation points.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

Simple layer potential method for domains having external corners

This note investigates the simple layer potential method for domains having external corners. The singular behaviour of simple layer density at the corners is studied for the Neumann problem of Helmholtz's equation. A numerical technique of solving the integral equation for this problem is proposed. This technique takes the singularity of the solution into consideration. Some numerical examples are given to show the applicability of the present method.     

Integral representation of a solution of the Neumann problem for the Stokes system

The Neumann problem for the Stokes system is studied on a domain in R ³ with Ljapunov bounded boundary. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series.     

Mixed Crack Type Problem in Anisotropic Elasticity

The paper deals with three-dimensional mixed boundary value problem of the anisotropic elasticity theory when the elastic body under consideration has a cut in the form of an arbitrary non-closed, two-dimensional, smooth surface with a smooth boundary: on one side of the cut surface the Dirichlet type condition (i.e., the displacement vector) is given, while on the other side the Neumann type condition (i.e., the stress vector) is prescribed. Applying the potential method and invoking the theory of ΨDEs uniqueness, existence and regularity results are proved in various function spaces. The asymptotic expansion of the solution of the corresponding system of boundary ΨDEs is written.     

On solvability of boundary integral equations of potential theory for a multidimensional cusp domain

The Dirichlet and the Neumann problems for the Laplace equation on a multidimensional cusp domain are considered. The unique solvability of the boundary integral equation for the internal Dirichlet problem for harmonic double layer potential is established. We also prove the unique solvability of the boundary integral equation for the external Neumann problem for harmonic single layer potential. Bibliography: 13 titles.     

A symmetric boundary element method for the Stokes problem in multiple connected domains

We consider a symmetric Galerkin boundary element method for the Stokes problem with general boundary conditions including slip conditions. The boundary value problem is reformulated as Steklov–Poincaré boundary integral equation which is then solved by a standard approximation scheme. An essential tool in our approach is the invertibility of the single layer potential which requires the definition of appropriate factor spaces due to the topology of the domain. Here we describe a modified boundary element approach to solve Dirichlet boundary value problems in multiple connected domains. A suitable extension of the standard single layer potential leads to an operator which is elliptic on the original function space. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation*

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered. The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied. An efficient algorithm for constructing the point itself a...     

Convergence of the piecewise linear approximation and collocation method for a hypersingular integral equation on a closed surface

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.     

An adapted Galerkin method for the resolution of Dirichlet and Neumann problems in a polygonal domain

The Neumann problem for Laplace's equation in a polygonal domain is associated with the exterior Dirichlet problem obtained by requiring the continuity of the potential through the boundary. Then the solution is the simple layer potential of the charge q on the boundary. q is the solution of a Fredholm integral equation of the second kind that we solve by the Galerkin method. The charge q has a singular part due to the corners, so the optimal order of convergence is not reached with a uniform mesh. We restore this optimal order by grading the mesh adequately near the corners. The interior Dirichlet problem is solved analogously, by expressing the solution as a double layer potential.