Research in Solvers for PDEs
One of the most important applications of numerical linear algebra is to the solution or approximate solution of a linear system of equations Ax=b. Indeed, this is one of the most fundamental problems in numerical analysis. When the linear system in question arises due to a discretisation of a partial differential equation (PDE) or a coupled system of PDEs, then the system matrix inherits many features from the underlying PDE operator and the discretisation methodology.
Solving such linear systems using naive elimination methods typically requires large computational memory resources and could involve many hours/days of run time. This is a serious bottle-neck in practice. Obtaining real time solutions on desktop or laptop computers is simply not possible in many applications without an optimised and dedicated solver exploiting our knowledge and understanding of the numerical discretisation methods and of the underlying PDE operators.
For very large problems, involving hundreds of thousands or millions of equations, iterative methods are the only viable approach. Usually, the convergence rate of such solvers depends on the conditioning and structure of the system matrix. These are heavily influenced by the properties of the underlying PDE operators. Preconditioning is then required. A preconditioner, in its simplest guise, can be thought of as a sparse matrix P that mimics the properties of the inverse of the original system matrix, but for which it is cheap to compute the matrix-vector product y=Pb. Designing so-called optimal preconditioners is a crucial step. Whilst some PDE operators such as the Laplacian are relatively `nice' and can be approximated using an algebraic or geometric multigrid cycle, others, such as the so-called H(div) or H(curl) operators are much more challenging to accommodate and are the subject of ongoing research.
In Manchester, we are currently carrying out research into a range of iterative methods and preconditioning techniques for solving PDEs that arise in diverse applications such as fluid flow modelling and electromagnetics. We have extensive experience in solving very large, sparse systems of equations that arise in finite element computations and particular expertise in solving saddle-point problems that arise in solving problems with constraints.
More recently, we have begun investigating robust solvers and preconditioners for PDEs with uncertain coefficients (for more details see uncertainty quantification). So-called stochastic finite element methods lead to linear systems that are orders of magnituge larger than for corresponding deterministic problems and their solution cannot be attempted without intelligent numerial linear algebra.