In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being rea...

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being rea...

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analy...

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in [52] for second order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions and, using certain discrete spaces an...

The computation of curvature quantities over discrete geometry is often required when processing geometry composed of meshes. Curvature information is often important for the purpose of shape analysis, feature recognition and geometry segmentation. In this paper we present a method for accurate estimation of curvature on discrete geometry especially those composed of meshes. We utilise a method...

We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [34] as well as in the case of a new fourth order scheme, which is discussed in this paper. It is based on the capacitance matrix method ([10], [8]). The discrete biharmonic operator is decomposed into two components. The fir...

A VLSI algorithm for solving a special block–five– diagonal system of linear algebraic equations will be presented. The algorithm is considered for the VLSI parallel computational model where both the time of the algorithm and the area of its design are components of the complexity estimations. The linear system arises from the finite– difference approximation of the first biharmonic boundary v...

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being written ...

Finite-volume discretizations can be formulated on unstructured meshes composed of different polygons. A staggered cell-vertex finite-volume discretization of shallow water equations is analyzed on mixed meshes composed of triangles and quads. Although triangular meshes are most flexible geometrically, quads are more efficient numerically and do not support spurious inertial modes of triangular...