Symmetric boundary knot method

The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general PDE’s. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a non-singular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, super-convergent, integration-free, very easy to learn and program. The original BKM, however, loses symmetricity in the presence of mixed boundary. In this study, by analogy with Fasshauer’s Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2-D and 3-D Helmholtz and diffusion-reaction problems under complicated geometries. Keyword: boundary knot method; radial basis function; meshfree; method of fundamental solution; dual reciprocity BEM; symmetricity * This research is supported by Norwegian Research Council. 1 1.Introduction In recent years much effort has been devoted to developing a variety of meshfree schemes for the numerical solution of partial differential equations (PDE’s). The driving force behind the scene is that the mesh-based methods such as the standard FEM and BEM often require prohibitive computational effort to mesh or remesh in handling highdimensional, moving boundary, and complex-shaped boundary problems. Many of the meshfree techniques available now are based on using the moving least square (MLS) strategy. In most cases, a shadow element is still necessary for numerical integration rather than for function interpolation. Therefore, these methods are not truly meshfree. Exceptionally, the methods based on the radial basis function (RBF) are inherently meshfree due to the fact that the RBF method does not employ the MLS at all and uses the one-dimensional distance variable irrespective of the dimensionality of the problems. Therefore, the RBF methods are independent of dimensionality and complexity of geometry. Nardini and Brebbia [1] in 1982 have actually applied the RBF concept to develop the currently popular dual reciprocity BEM (DR-BEM) without the notion of “RBF” and the use of then the related advances in multivariate scattered data processing. Only after Kansa’s pioneering work [2] in 1990, the research on the RBF method for PDE’s has become very active. Among the existing RBF schemes, the so-called Kansa’s method is a domain-type collocation technique, while the method of fundamental solution (MFS) (regular BEM versus singular BEM) is a typical boundary-type RBF collocation methodology. The MFS outperforms the standard BEM in terms of integration-free, super-convergent, easyto-use, and meshfree merits [3]. The main drawback of the MFS is the use of fictitious boundaries outside the physical domain. The arbitrariness in the determination of the artificial boundary introduces such troublesome issues as stability and accuracy in dealing with complicated geometry systems and undercuts the efficiency of the MFS to practical engineering problems [4,5].