# Overview

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## Solving large systems efficiently in multiphysics numerical simulations - May 31-June 10, 2021

Multiphysics simulations have become an important problem that has practical applications in many fields in science and engineering. Two examples include (i) simulating the interactions between the flow of fluids in the subsurface (e.g. groundwater, oil and natural gas) and the mechanical deformation of the surrounding rock due to pressure, temperature, etc.; (ii) simulating the interactions between the ocean and the atmosphere in meteorological predictions and climate modelling. In these and many other applications, the underlying model in each regime is given by partial differential equations (PDEs), which are then discretized and solved many times over the course of the simulation. The resulting algebraic systems are typically large (with millions of degrees of freedom), sparse (each equation only depends on a few unknowns) and ill-conditioned (very sensitive to perturbations in the data), which makes their efficient solution challenging. The past 50 years have seen the rise of iterative methods and preconditioners that are highly efficient for particular types of PDEs, such as heat diffusion, fluid flow, waves arising from electromagnetism, etc. However, when different physical models interact, it becomes essential to develop techniques to solve the coupled problem efficiently, in addition to the individual components. The goal of this summer school is to introduce fundamental solver techniques for dealing with these coupled problems. In particular, we are interested in techniques that are able to exploit modern supercomputing architecture, which often have tens of thousands of processors capable of doing many calculations in parallel.

The theoretical portion of the summer school consists of three mini-courses that introduce the participants to fundamental techniques for large multiphysics problems: (i) stationary iterative methods, (ii) domain decomposition, and (iii) multigrid methods. These mini-courses will be complemented by practical sessions, where participants will be guided by well-designed problem sets in order to experiment with the methods introduced in the mini-courses, and to explore the mathematics behind these techniques. In addition there will be four seminar-style talks by invited speakers, who will present their work on multiphysics simulations and/or preconditioning techniques.

**The school is free of charge, but registration is mandatory. Due to limited capacity, we can only accept a maximum of 200 participants on a first-come-first-serve basis..**