### Numerical methods for Stokes flow

Stokes flow arises in many applications, including modelling ice sheets, lava and polymers. This type of viscous flow is described by the Stokes equations, which couple together the fluid’s velocity and pressure. Often, fine resolution solutions of the Stokes equations are required in applications, the computation of which requires sophisticated numerical methods.

This course will discuss how to efficiently solve Stokes problems from start to finish. We will look at the wellposedness of these equations and their discretisation by a Galerkin finite element method. A key component of finite element (and other) discretisations is the solution of a system of equations with a large, sparse coefficient matrix. These linear systems are typically solved by preconditioned Krylov subspace methods, a family of iterative methods that exploit sparsity.

The techniques learnt here can be applied to other differential equations, and other linear systems of equations.

### Course Outline

1. Stokes flow: motivation and inf-sup conditions
We will introduce and motivate the Stokes equations before deriving the weak form. This will enable us to relate the Stokes equations to a constrained optimisation problem that leads naturally to the inf-sup condition for wellposedness of Stokes problems.
2. The finite element method for Stokes problems
We will start by deriving the finite element method for a Poisson problem, before developing mixed finite element methods for the Stokes equations. A key issue, on which we will focus, is understanding when a discrete inf-sup condition holds.
3. Iterative methods for the Stokes equations
To obtain the finite element solution to the Stokes equations we must solve a linear system. The final part of this course investigates how to do this using Krylov subspace methods. We will use the MINRES method, and will look at how to accelerate its convergence rate by preconditioning.

### Pre-requisites

The lectures require basic knowledge of linear algebra and partial differential equations. Proficiency in MATLAB is also required.

### Pre-reading & Lecture Notes

Access lecture notes for this course

Pre-reading

A Finite Element Primer” by D. J. Silvester in advance of the course: https://personalpages.manchester.ac.uk/staff/david.silvester/primer.pdf

A good reference for this course is the book:
H. Elman, D. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, UK, 2014. (We will look at material from Chapters 1, 3 and 4.) #### Dr Jen Pestana, University of Strathclyde, Glasgow

Jen Pestana is a lecturer in numerical analysis at the University of Strathclyde. Prior to this she received her DPhil in Numerical Analysis from Oxford and held postdoctoral positions at The University of Manchester and Oxford University.

Jen’s research lies in the field of numerical linear algebra, a branch of numerical analysis that deals with matrix problems. Most of her work is on the development and analysis of tailored iterative methods for solving linear systems arising from scientific computing applications. She is particularly interested in understanding and exploiting a problems’s structure, and the relationship between a linear system of equations and the underlying (usually continuous) problem. She also conducts research into characterising the convergence rate of iterative methods for nonsymmetric (non-self-adjoint) problems.