## PhD research topic - Maria Lopez Fernandez

#### Research areas

Numerical analysis of evolutionary problems, numerical complex analysis

#### Keywords

Wave scattering problems, boundary integral equations, convolution equations, retarded potentials, fractional differential equations, contour integral methods, adaptivity, fast algorithms

#### Research topics

The efficient resolution of exterior wave problems, both acoustic and electromagnetic, have become an active field of research in the last decade. A well established approach to deal with the possibly unboundedness of the computational domain in the reformulation of the problems as time-domain boundary integral equations. In this way the computational domain becomes the bounded surface of the scatterers. The main difficulties arising in the space-time discretizazion of the resulting retarded integral equations are the numerical stability, the computational cost and the memory requirements. Several strategies have been proposed to overcome these bottlenecks and are currently under study: introduction of time-space adaptivity, hierarchical matrix techniques to deal with associated high-frequency problems, convolution quadrature methods, etc.

#### Bibliography

[1] L. Banjai, M. L\'opez-Fern\'andez, A. Sch\"adle, Fast and oblivious convolution quadrature for dissipative and two-dimensional wave equations, SIAM Journal on Numerical Analysis 55 (2017), pp.~621--639.

[2] M. Lopez-Fernandez and S. Sauter, Generalized Convolution Quadrature with Variable Time Stepping, IMA J. Numer. Anal. 33 (2013), 1156-1175.

[3] F.-J. Sayas, Retarded Potentials and Time Domain Boundary Integral Equations. Springer Series in Computational Mathematics, 50, 2016.