Modern numerical methods for partial differential equations

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Partial differential equations (PDEs) are among the most important objects of study in science and engineering. Solution of these equations would reveal the properties of the system under study. Very often, these equations express conservations of mass, energy and momenta. For instance, aircraft manufacturers are interested in designing wings of aircraft maximizing efficiency, performance and safety. The relevant PDEs are the conservation laws mentioned above for the unknown velocity and pressure fields. Unfortunately, explicit solutions of these equations are very rare. In general scientists and engineers rely on numerical methods to solve PDEs. In realistic 3D cases, there may be many millions of unknowns and equations. The most powerful computers are not able to solve these equations in reasonable time using classical methods.***Spectral methods are classical numerical methods to solve PDEs. If the PDE is time independent and the solution is smooth, then spectral methods converge exponentially, meaning that for the same order of accuracy, only thousands of unknowns are required compared to many millions of unknowns for other methods which don't converge exponentially. Hence spectral methods can obtain the solution much quicker compared to other methods. Unfortunately, if the PDE is time dependent, the classical spectral method does not converge exponentially. Space-time spectral methods are new methods which do converge exponentially and have appeared only within the past decade.***In the current Discovery cycle, I have proven exponential convergence of a space-time spectral method for the heat equation, a time dependent PDE of great significance which describes the temperature distribution of a body. Together with my students, the next step is to repeat for other standard linear PDEs occurring in science and engineering. After that, nonlinear equations can be considered. Another important aspect is to design fast solvers for the equations which arise in the space-time spectral method. Finally, the classical spectral method works only for rectangular geometry. For problems on complex geometry, the spectral element method, a generalization of the classical spectral discretization, can be considered.***Another topic to be explored involves numerical methods for fractional PDEs. Classical PDEs model local phenomena, while fractional PDEs are for systems exhibiting nonlocal behaviour. Fractional PDEs have been studied mostly in the past decade and have been one of the most active areas of mathematics. Numerical methods for fractional PDEs have only appeared in the past five years.***Very recently, I have performed a convergence analysis of a finite difference method for a 1D time independent fractional equation. Some of the aims of this program include extending the analysis to higher dimensions and time dependent problems, and designing modern fast solvers for the resultant equations.**