Felix Kwok Section de mathématiques Université de Genève 2–4 rue du Lièvre, CP 64 1211 Genève 4, Switzerland

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Research interests

My general research interests revolve around algorithms and techniques for efficient large-scale simulations in physics and engineering, particularly when the models are described by PDEs or systems of PDEs. Typically, such simulations must repeatedly solve large, sparse linear and nonlinear systems of equations that result from discretizing the PDEs. To extract maximum efficiency, it is crucial to employ fast, robust and mathematically sound linear and nonlinear solvers, specialized to exploit both the underlying physics of the problem and well-known numerical techniques. Solver design is thus a fascinating topic that lies at the crossroads of mathematical analysis, scientific computing and engineering. It is this fundamentally interdisciplinary nature that I find particularly stimulating and rewarding.

Current projects

Domain Decomposition Methods

In domain decomposition methods, the computational domain is divided into several subdomains, and the subdomain equations are solved in parallel. An iteration is then performed to ensure the solutions are consistent across subdomain boundaries. The convergence rate of this iteration is highly dependent on which boundary conditions are used along subdomain boundaries. Our goal is to derive optimized transmission conditions algebraically. We have shown that with a suitable (nonlocal) operator, it is possible to achieve convergence in a finite number of steps, regardless of how the subdomains are interconnected. Ongoing work focuses on the local approximation of such operators and how they affect the asymptotic convergence rate under grid refinement. Slides on optimal interface conditions, presented at the 19th Domain Decomposition Conference in Zhangjiajie, China (PDF, 397kb) Slides on equivalence between ASH and parallel Schwarz methods, presented at the 19th Domain Decomposition Conference in Zhangjiajie, China (PDF, 501kb)

Fast solvers for reservoir simulation

The reservoir equations, though highly nonlinear, are directional in nature because it models fluid flow in a porous medium. Linear and nonlinear solvers can exploit this directionality to obtain a partial decoupling of the unknowns and solve them sequentially. The resulting Newton-based nonlinear solver becomes more robust with respect to time-step sizes. Meanwhile, the linear solver also converges more quickly and becomes less sensitive to flow configurations. Ongoing research seeks to quantify these benefits and to extend the methodology to other problems that are directional in nature, such as Navier-Stokes. Slides on ordering methods in reservoir simulation (PDF, 1.6Mb)

Fully-Implicit Method Implicit Pressure/Explicit Saturation