Let T be a tree rooted at e endowed with a nearest-neighbor transition probability that yields a recurrent random walk. We show that there exists a function K biharmonic off e whose Laplacian has potential theoretic importance and, in addition, has the following property: Any function f on T which is biharmonic outside a finite set has a representation, unique up to addition of a harmonic funct...

During the development of a convergence theory for Nicolaides’ extension [21, 24] of the classical MAC scheme [25, 22, 26] for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MA...

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the ...

SUMMARY Applying a finite difference approximation to a biharmonic equation results in a very ill-conditioned system of equations. This paper examines the conjugate gradient method ' used in con junction with the generalized and approximate polynomial preconditionings for solving such linear systems. An approximate polynomial preconditioning is introduced, and is shown to be more efficient than...

In this paper, getting an high-efficiency parallel algorithm to solve sparse block pentadiagonal linear systems suitable for vectors and parallel processors, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are appropriate when the desired target is to maximize parallelism. Moreover, some theoretical results about these pre...