Meshfree methods for parabolic equations based on weighted least-squares

This project considers kernel-based meshfree methods for solving parabolic partial dif- ferential equations. For time-independent problems, these are widely used and well un- derstood, but this project plans to extend them to time-dependent “parabolic” problems that arise in Science and Engineering whenever diffusion or dispersion effects come into play, e.g. in air or water pollution or heat flux. We plan to develop novel numeri- cal methods and provide a rigid theoretical analysis for them. After defining a general framework for the standard (non-nodal) and nodal notions of kernel-based trial spaces that have proven to be very useful in time-independent settings, we first show that there exist comparison functions in trial spaces that are very good approximations to the true solution of the problem. Their convergence rates are determined by the employed kernel and the smoothness of the solution but not the parabolic partial differential equation. The next challenging task is to come up with numerically stable methods to calculate a trial function from the given data that converges as fast as the comparison function.

Our methods will apply collocation and overtesting, because there are plenty of ap- plications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size but is also known to be necessary for stability and convergence of widely used unsymmetric Kansa- type strong-form collocation methods. The overtesting technique has not yet been applied to parabolic partial differential equations.

Via overtested strong collocation, we can obtain various semi-discretized solutions of parabolic partial differential equations, in which the time variable remains continuous. For additional stability and for reducing computational complexity, we shall focus on least-squares methods to handle the overtesting numerically. Using the heat equation as an example, we want to show that overtesting yields a very good approximation of the spectrum of the analytical non-discretized evolution operator. In particular, preliminary numerical tests showed that the numerical performance of a weighted least-squares tech- nique for handling overtesting combined with a sensible choice of trial spaces stands out from competing standard methods, and it is the main objective of this project to provide a mathematical foundation for this observation.

Summarizing, the purpose of this project is to provide a new class of numerical tech- niques for solving parabolic partial differential equations and to establish a solid mathe- matical foundation for it.