Asymptotically linear elliptic boundary value problems

Factorization of linear elliptic boundary value problems in non-cylindrical domains

The Galerkin finite elements method (GAFEM) is used to find the numerical solution (NS) of the couple linear elliptic boundary value problem (CLEBVP). The solution of the CLEBVP is approximated by a linear combination of piecewise linear and continuous basis functions, upon substituting this approximated solution into the CLEBVP the problem then reduces to solve a “Galerkin” linear algebraic system (GALAS). This system is solved by using the classical Cholesky method (CChM). Some illustrative examples are given; to show the high efficiency of the GAFEM to solve such for the CLEBVPs. Figures provide the NS with “results”.

Morse–Smale index theorems for elliptic boundary deformation problems

Chapter 7 Preconditioned Iterative Methods for Nonselfadjoint or Indefinite Elliptic Boundary Value Problems

This paper investigates the Mikhlin stability of the Rayleigh-Ritz method for certain classes of linear elliptic boundary value problems. Piecewise polynomial coordinate functions are studied from the point of view of Mikhlin stability, and in particular, it is shown that appropriately scaled B-splines and appropriately scaled elementary Hermites yield a Mikhlin stable Rayleigh-Ritz-Galerkin process and approximate solution for the classes of problems considered. Firstly, a class of two-point boundary value problems studied by Birkhoff et al. (1968) is investigated, and then the results are extended to higher dimensions.

A generalized monotonicity theorem in the sense of Collatz is proved for almost linear elliptic boundary value problems of second order. This theorem is valid for continuous spline functions as used by the finite element method — in contrast to the classical theorems. The background is a generalized maximum principle with an interface condition for the normal derivatives which has been proven by Natterer/Werner in the case of the Laplace operator. Some numerical examples providing pointwise bounds for the solution of Dirichlet problems by cubic splines, the so called T 10 elements, are given.

In this paper, an h p hp -local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On h p hp -quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H 1 H^1 norm and L 2 L^2 norm which are optimal in h h , suboptimal in p p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

We consider estimates depending on a parameter for general linear elliptic boundary value problems, with nonhomogeneous boundary conditions, in Nikol’skij spaces. These estimates are then employed to study general linear nonautonomous parabolic systems, again with nonhomogeneous boundary conditions. Maximal regularity results are proved.

Consideration of an abstract improvement algorithm leads to the following principle, which is similar to that underlying iterative refinement: By making judicious use of relatively few high accuracy computations, high accuracy solutions can be obtained very efficiently by the algorithm. This principle is applied specifically to ${\text{GMRES}}(m)$ here; it can be similarly applied to a number of other “restarted” iterative linear methods as well. Results are given for numerical experiments in solving a discretized linear elliptic boundary value problem and in computing a step of an inexact Newton method using finite differences for a discretized nonlinear elliptic boundary value problem.

The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier--Stokes equations, $N \leq 4$. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small.

Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.

Fast iterative solution of elliptic control problems in wavelet discretization

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the computing procedures to verify the existence, uniqueness and inclusion set of a solution based on Schauder’s fixed point theorem. Using the finite element approximations for some simple Poisson’s equations and the results of error estimates, we generate iteratively a set sequence composed of functions and attempt to construct automatically the set including the exact solution. Further, the conditions of verifiability by this method are considered and some numerical examples of verification are presented.

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Weighted analytic regularity in polyhedra