# Catherine E. Powell's Research Page

My research is mainly concerned with aspects of numerical analysis related to the numerical solution of partial differential equations (PDEs). In particular, I am interested in the numerical solution of PDEs with uncertain/random inputs in engineering applications and efficient and accurate numerical algorithms for uncertainty quantification (UQ).

Key Topics: numerical analysis, UQ, finite elements, mixed finite elements, error estimation, a posteriori error estimation, adaptivity, stochastic finite elements, numerical linear algebra, saddlepoint systems, fast solvers, iterative solvers, preconditioning.

### Textbook

• An Introduction to Computational Stochastic PDEs (by G. Lord, C.E. Powell and T. Shardlow). Preface.
• Textbook designed to introduce recent graduates with a good grounding in applied mathematics and numerical analysis to stochastic differential equations.
• Assumes no prior knowledge of probability or statistics.
• MATLAB and python codes associated with the book are available HERE. Solutions to exercises are available for verified course instructions (by emailing one of the authors).

### Software

• IFISS3D: An add-on MATLAB toolbox that extends the IFISS (Incompressible Flow & Iterative Solver) finite element software package for (deterministic) PDEs to three dimensional spatial domains. This code is intended to be useful for teaching purposes and as a source of reproducible test matrices of arbitrarily large dimension arising from finite element discretisations.

• ML-SGFEM: A MATLAB toolbox for investigating adaptive multilevel stochastic Galerkin finite element approximation of parametric elliptic PDEs, driven by hierarchical error estimation. This code is associated with the paper: SIAM J. Comp. Sci., 41(3), A1681--A1705, (2019). A User Guide is HERE.

• S-IFISS: a simple extension of the IFISS software that implements (single level) stochastic Galerkin finite element discretisations of parametric elliptic PDEs. This code is also provided as supplementary material for the a posteriori error estimation strategy described in the paper: SIAM Journal Sci. Comp. 36(2), A339--A363 (2014).

• PIFISS: (Potential Incompressible Flow Software Library). This code illustrates the use of Krylov subspace solvers and preconditioning techniques based on algebraic multigrid for primal and mixed formulations of the steady-state diffusion problem. Test problems are included with discontinuous and anisotropic diffusion coefficients.

• Gaussian random field generators in the NAG Fortran Library. The routines G05ZMF, G05ZNF, G05ZPF G05ZQF, G05ZRF, G05ZSF and G05ZTF in Mark 24 of the NAG Fortran Library are based on the method of circulant embedding. These were developed during a collaboration between PhD student Phillip Taylor, Tony Shardlow and NAG (who partially funded the PhD project). The codes can be used to generate realisations of mean zero Gaussian random fields and stochastic processes with both standard and user-defined stationary covariance functions. Technical report in NAG news 119 , January 2014.