Solving biharmonic equation using the localized method of approximate particular solutions

Abstract

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.

Local RBF Algorithms for Elliptic Boundary Value Problems in Annular Domains

“…The difficulties encountered when local methods are used to solve biharmonic problems are well-document in the literature [3,24]. Hence, the relatively low accuracy of the results obtained in this example is not unusual.…”

“…For the Laplacian boundary condition, the local formulation is practically the same as (4.7)-(4.9). Note that in the second biharmonic problem, the problem may be decoupled into two Poisson problems which could be subsequently solved by the method described in Section 4.1, see [24].…”

“…Improved results can be obtained when the second biharmonic problem consisting of (2.3a) and (2.3c) is split into two Poisson Dirichlet problems and these are subsequently solved using Algorithm 1, described in Section 4.2. Further details regarding this approach may be found in [24].…”