mscroggs.co.uk
mscroggs.co.uk

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I am currently a PhD student at University College London (UCL) working on boundary element methods under the supervision of Timo Betcke and Erik Burman.

### Bempp

During my PhD, I have become heavily involved in developing Bempp, an open-source boundary element method library.
Bempp is primarily written in Python with a fast C++ computational core. Documentation can be found at bempp.com.

### Maxwell wave scattering

An electromagnetic wave scattering off
a birthday cake.
During my PhD, I have done much work on boundary element methods for Maxwell wave scattering problems. In Bempp, I implemented Buffa–Christiansen dual function spaces: these spaces are required when multiplying two operators together which is necessary when applying Calderón preconditioning to the electric field integral equation (EFIE).
These spaces are also useful when stably discretising the magnetic (MFIE) and combined (CFIE) field integral equations. This work is discussed in more detail in Software frameworks for integral equations in electromagnetic scattering based on Calderón identities (2017).

### Weak imposition of boundary conditions

More recently, I have been working on a method of weakly imposing boundary conditions when using the boundary element method. This method weakly imposes the boundary conditions by adding a penalty term to the full Calderón system, inspired by Nitsche's method for finite element methods.
Due to the approximate double of the number of unknowns, this method is not competitive for pure Dirichlet and Neumann problems, but for more complex boundary conditions—such as Robin boundary conditions and mixed boundary conditions—it provides a much simpler formulation that is easier to implement.
We analysed this method for Laplace Dirichlet, Neumann and Robin problems in Boundary element methods with weakly imposed boundary conditions (2018), and are currently working on an application of it to the Signorini problem.

### FEM-BEM coupling

The solution of a coupled FEM-BEM
Helmholtz problem.
Early in my PhD, I worked on the coupling of the finite and boundary element methods. For transmission problems involving wave travelling through a large or infinite medium and through a small inhomogeneous object, the boundary element method is well suited to solving the problem outside the object while the finite element method is better suited for solving the problem inside the object. Using the two methods for the two parts of the problem gives rise to FEM-BEM coupling.
A tutorial showing how to implement FEM-BEM coupling using Bempp and FEniCS can be found here.

A full list of my papers, conference talks, awards and nominations can be found below.
2018

#### Papers

T. Betcke, M. W. Scroggs & W. Śmigaj. Product algebras for Galerkin discretisations of boundary integral operators and their applications, submitted to ACM Transactions on Mathematical Software. [arΧiv]
T. Betcke, E. Burman & M. W. Scroggs. Boundary element methods with weakly imposed boundary conditions, submitted to SIAM Journal on Scientific Computing. [arΧiv]
A. Kleanthous, T. Betcke, D. P. Hewett, M. W. Scroggs & A. J. Baran. Well-conditioned boundary integral equations for multiple scattering by absorbing dielectric particles, submitted to Journal of Quantitative Spectroscopy and Radiative Transfer. [arΧiv]

#### Conference talks

Solving integral equations for electromagnetic scattering using Bempp (poster) (PDESoft 2018)
Weak imposition of boundary conditions using a penalty method (IABEM 2018)
Weak imposition of boundary conditions using a penalty method (Söllerhaus Workshop on Fast Boundary Element Methods in Industrial Applications)
2017

#### Papers

M. W. Scroggs, T. Betcke, E. Burman, W. Śmigaj & E. van 't Wout. Software frameworks for integral equations in electromagnetic scattering based on Calderón identities, Computers & Mathematics with Applications 74(11), 2897–2914. [arΧiv, doi.org/10.1016/j.camwa.2017.07.049]

#### Conference talks

Solving integral equations for electromagnetic scattering using BEM++ (Strathclyde 27th Biennial Numerical Analysis Conference)
Solving integral equations for electromagnetic scattering using BEM++ (Söllerhaus Workshop on Fast Boundary Element Methods in Industrial Applications)
2016

#### Conference talks

Solving Maxwell problems with BEM++ (Joint DMV and GAMM Annual Meeting)
Coupling the finite and boundary element methods with FEniCS and BEM++ (BAMC 2016)
FEM-BEM coupling, Maxwell's equations, and BEM++ (FEniCS'16)
Coupling the finite and boundary element methods with FEniCS and BEM++ (PDESoft 2016)

#### Teaching

In 2015-16 and 2016-17, I taught MATH6103: Differential and Integral Calculus. You can find my notes for this course at mscroggs.co.uk/6103.
2015

#### Conference talks

Solving FEM/BEM coupled problems with FEniCS and BEM++ (Strathclyde 26th Biennial Numerical Analysis Conference)
Using FEniCS with BEM++ For FEM/BEM Coupled Problems (FEniCS'15) [video]