Error-Optimal Methods for PDEs Arise from Meshless Theories

Many problems in science and engineering require solving partial differential equations (PDEs). The area of meshless methods, including radial basis function (RBF) collocation or the Kansa’s method, is an important area of contemporary research into numerical PDEs; most works focus on elliptic and parabolic types. Meshless methods enjoy an advantages that they do not require construction and possible manipulation of grids. Hence, the rather short history of meshless methods already features plenty of nicely calculated practical applications. On the other hand, the theories of the Kansa’s method are very limited. Some researchers may not be persuaded that the Kansa’s method is the right approach at all.

Providing any theoretical result for the Kansa’s method, we look at the issue that motivated Kansa back then, the RBF interpolation problem which has far more complete theoretical supports, and use the generic error estimates for (generalized) RBF interpo- lation to derive new numerical methods for solving PDEs. To be more specific, a generic error estimate of RBF interpolation (analogous to that for polynomial interpolations) says that the (L∞) difference between the RBF interpolant and the function is bounded by the (Reproducing Kernel Hilbert Space, RKHS) norm of the function times some constant independent of the function. This constant, commonly known as the power function, can then be further bounded in terms of the fill distance of data to complete the error analysis.

In this project, we focus on utilizing the power function, which depends on both the RKHS, and the distribution of the data, to design new algorithms. Once the kernel and hence its RKHS are fixed, the function or PDE solution and its norm are also specified. Moreover, using the Mat ́ern kernels, which reproduce the standard Sobolev spaces, we are able to evaluate error (up to a factor of the norm of the solution) in the familiar Sobolev sense, rather than some abstract RKHS. Based on information extracted from the corresponding power function, we can design algorithms to minimize the power function (over some Sobolev norms) under various constraints for the sake of numerical efficiency; hence, we title this project as Error-Optimal methods.