Partial differential equations (PDEs) and functional differential equations (FDEs) are successful, widely used mathematical models that describe, explain and predict phenomena in areas as broad as physics, chemistry, biology and economics. However, since these equations are naturally defined on infinite dimensional function spaces, the beautiful and highly complex patterns generated by their solutions are often difficult to capture using standard analytic techniques. As a result of these difficulties and with the availability of powerful computers and sophisticated software, numerical simulations are often the primary tool used by scientists and engineers confronted with these infinite dimensional problems. The computational methods used in simulations require a finite dimensional, bounded, discrete setting, whereas solutions of PDEs and FDEs are infinite dimensional, unbounded and continuous. With any numerical method there is the question of reliability of the output. How can one indeed make sure that the truncation error term involved in computing on a finite dimensional approximation will not lead to spurious solutions? How to control the appearing computational errors? The goal of this proposal is to address the above questions and perform the efficient development and implementation of reliable computational methods that capture and imitate the most essential features, and also control errors for approximations of the solutions of the above mentioned infinite dimensional problems. The research program proposed will consist of the following four general themes:

  1. Enchancing the reliability of numerical simulations for PDEs
  2. Implementation of algorithms and procedures proposed in A
  3. Computer-assisted proofs for nonlinear PDEs
  4. Computer-assisted proofs for FDEs
In A and B we will theoretically develop new computational tools for enchancing reliability of computer simulations for PDE models and perform their efficient implementation on the modern supercomputers with parallel architecture. In C, we will develop computational methods to prove existence of solutions of nonlinear PDEs, with a special emphasis on the Navier-Stokes equations, and in D, we will develop rigorous computational methods for proving the existence of certain solutions of functional differential equations.