Research

Time-domain acoustic and electromagentic wave propagation

Time-domain boundary integral equations

The analysis and fast implementation of convolution quadrature applied to boundary integral equations of acoustic and electromagentic scattering. Examples of works in this direction:

• Banjai L., Melenk J.M., and Lubich Ch.: Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math., 119(1), 1–20, 2011.

• Banjai L.: Multistep and multistage convolution quadrature for the wave equation: Algorithms and experiments, SIAM J. Sci. Comput., 32(5), 2964–2994, 2010.

• Ballani, J., Banjai, L., Sauter, S., and Veit, A., Numerical Solution of Exterior Maxwell Problems by Galerkin BEM and Runge-Kutta Convolution Quadrature, Numer. Math., 123(4), 643–670, 2013.

• Banjai, L., Lubich, Ch., and Sayas, F.J., Analysis of FEM-BEM coupling in time domain , accepted for publication in Numerische Mathematik

• Banjai, L. and Kachanovska, M., Fast convolution quadrature for the wave equation in three dimensions, accepted for publication in JCOMP

Space-time Trefftz methods

We investigate the use of the discrete spaces of local solution to the wave equations as approximation spaces for transient wave propagation. We have developed a new space-time interior penalty discontinuous Galerkin method that incorporates these spaces.

1. Banjai, L., Georgoulis, E.H. and Lijoka, O. A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation, accepted for publication in SINUM, 2016 arXiv:1610.01878.

Parallel solution of evolution problems

Here we investigate the solution in parallel of time stepping schemes for parabolic and hyperbolic linear equations.

• Banjai L. and Peterseim, D.: Parallel multistep methods for linear evolution problems, IMA J. Numer. Anal., 32, 1217–1240, 2012.

Frequency domain wave propagation

Wave number dependence of numerical schemes

Here we have investigated the dependence of the error and stability of FEM and BEM discretizations of the Helmholtz equation at high frequences:

• Banjai, L. and Sauter, S. A refined Galerkin error and stability analysis for highly indefinite variational problems, SIAM J. on Num. Anal., 45(1), 37–53, 2007.

Data sparse methods for integral equations

I have use fast multipole methods (FMM) and hierarchical matrices for boundary integral equations for frequency and time domain computations. Further I have applied to integral equations stemming from computational complex analysis. Relevant papers are:

• Banjai, L. and Kachanovska, M., Fast convolution quadrature for the wave equation in three dimensions, submitted to JCOMP

• Banjai, L. and Hackbusch W. Hierarchical matrix techniques for low and high frequency Helmholtz problems, IMA J. Numer. Anal.,28, 46–79, 2008.

• Banjai, L. and Trefethen, L. N. A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides. SIAM J. Sci. Comp. 25(3), 1042-1065, 2003.

Eigenvalue computations

I have computed high accuracy approximations of eigenmodes of fractal drums. The techniques used were conformal transplantation and spectral methods on the unit disk. Further, I have been involved in the investigatation of the requirements on the mesh size for multigrid accelarated solution of eigenvalue problems.

• Banjai, L. Eigenfrequencies of fractal drums. J. Comput. Appl. Math. 198(1), 1-18, 2007.

• Banjai, L., Boerm, S., and Sauter, S. FEM for Elliptic Eigenvalue Problems: How Coarse Can the Coarsest Mesh be Chosen? An Experimental Study., Comp. and Visualisation in Science, 11, no. 4-6, 363–372, 2008.

Computational complex analysis

• Banjai, L. and Trefethen L. N. Numerical solution of the omitted area problem of univalent function theory. Comput. Methods Funct. Theory 1(1), 259-267, 2001.

• Banjai, L. and Trefethen, L. N. A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides. SIAM J. Sci. Comp. 25(3), 1042-1065, 2003.