The Augmented Scattering Matrix and Exponentially Decaying Solutions of an Elliptic Problem in a Cylindrical Domain

The self-adjoint elliptic boundary-value problem in a domain with cylindrical outlets to infinity is considered. The notion of an augmented scattering matrix is introduced on the basis of artificial radiation conditions. Properties of the augmented scattering matrix are studied, and the relationship with the classical scattering matrix is demonstrated. The central point is the possibility of calculating the number of linearly independent solutions of a homogeneous problem with fixed rate of decrease at infinity by analyzing the spectrum of the augmented scattering matrix. This property is applied to the problem on diffraction on a periodic boundary as an example. Bibliography: 21 titles.     

A discontinuous Galerkin method for Stokes equation by divergence-free patch reconstruction

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle-point problem due to such weak form. The number of degree of freedoms of our method is the same as the number of elements in the mesh for different order of accuracy. The error estimations of the proposed method are given in a classical style, which are then verified by some numerical examples.     

On a Toeplitz Representation of the H1/2 Norm

Some representations of the H1/2 norm are used as Schur complement preconditioner in PCG based domain decomposition algorithms for elliptic problems. These norm representations are efficient preconditioners but the corresponding matrices are dense, so they need FFT algorithm for matrix-vector multiplications. Here we give a new matrix representation of this norm by a special Toeplitz matrix. It contains only O(log(n)) different entries at each row, where n is the number of rows and so a matrix-vector computation can be done by O(nlog(n)) arithmetic operation without using FFT algorithm. The special properties of this matrix assure that it can be used as preconditioner. This is proved by estimating spectral equivalence constants and this fact has also been verified by numerical tests.     

Optimal interior partial regularity¶for nonlinear elliptic systems: the method of A-harmonic approximation

We consider nonlinear elliptic systems of divergence type. We provide a new method for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. This method is applied to both homogeneous and inhomogeneous systems, in the latter case with inhomogeneity obeying the natural growth condition. Our methods extend previous partial regularity results, directly establishing the optimal H?lder exponent for the derivative of a weak solution on its regular set. We also indicate how the technique can be applied to further simplify the proof of partial regularity for quasilinear elliptic systems. Received: 22 July 1999 / Revised version: 23 May 2000     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

On the convergence of a dual-primal substructuring method

Summary.   In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000     

Green's functions for elliptic and parabolic systems with Robin-type boundary conditions

The aim of this paper is to investigate Green's function for parabolic and elliptic systems satisfying a possibly nonlocal Robin-type boundary condition. We construct Green's function for parabolic systems with time-dependent coefficients satisfying a possibly nonlocal Robin-type boundary condition assuming that weak solutions of the system are locally Hölder continuous in the interior of the domain, and as a corollary we construct Green's function for elliptic system with a Robin-type condition. Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary. We provide some examples satisfying such a local boundedness property, and thus have Gaussian bounds for their Green's functions.     

Lyapunov diagonal semistability of acyclic matrices

It is shown that an acyclic matrix is Lyapunov diagonally semistable if and only if the matrix has the weak principal submatrix rank property. This result completes the solution of the problem of characterizing the various types of matrix stability for acyclic matrices. Also, those acyclic matrices which have a unique Lyapunov scaling factor are characterized.     

Regularity of Solutions to Problems for Nondiagonal Quasilinear Elliptic Systems with Conditions on Interface Surfaces

We consider a model problem for a quasilinear elliptic system with nondiagonal principal matrix and certain conditions on a surface dividing a medium. The partial regularity of a weak solution in a neighborhood of the interface is established. Bibliography: 11 titles.     

Convergence of the ghost fluid method for elliptic equations with interfaces

This paper proves the convergence of the ghost fluid method for second order elliptic partial differential equations with interfacial jumps. A weak formulation of the problem is first presented, which then yields the existence and uniqueness of a solution to the problem by classical methods. It is shown that the application of the ghost fluid method by Fedkiw, Kang, and Liu to this problem can be obtained in a natural way through discretization of the weak formulation. An abstract framework is given for proving the convergence of finite difference methods derived from a weak problem, and as a consequence, the ghost fluid method is proved to be convergent.