In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invoke...

The main result of this paper is the construction of a conformally covariant operator in two dimensions acting on scalar fields and containing fourth order derivatives. In this way it is possible to derive a class of Lagrangians invariant under conformal transformations. They define conformal field theories satisfying equations of the biharmonic type. Two kinds of these biharmonic field theorie...

In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base f...

The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

Recently the biharmonic form of the Navier-Stokes (N-S) equations have been solved in various domains by using second order compact discretization. In this paper, we present a fourth order essentially compact (4OEC) finite difference scheme for the steady N-S equations in geometries beyond rectangular. As a further advancement to the earlier formulations on the classical biharmonic equation tha...

Among various barycentric coordinates and their extensions, the linear and cubic (Hermite) Gordon-Wixom transfinite interpolation schemes deliver the most accurate approximations of the harmonic and biharmonic functions, respectively. However interpolation properties of the original Gordon-Wixom interpolations are studied for convex domains only and, therefore, their current practical importanc...

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...

This paper is concerned with weak solution of a mixed boundary value problem for the biharmonic equation in the plane. Using Green’s formula, the problem is converted into a system of Fredholm integral equations for the unknown data on different part of the boundary. Existence and uniqueness of the solutions of the system of boundary integral equations are established in appropriate Sobolev spa...

In this paper, we investigate the existence of infinitely many solutions for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms. The solutions are obtained from a variant version of Fountain Theorem.