Personal research interests

You can find a brief information on the research group of Karol Mikula here.

Currently I am mostly interested in the enhancement and applications of level set methods to model evolving interfaces and surfaces. Additionally during my stays at universities in  Erlangen (Germany, 1995-1998), Ghent (Belgium, 1998-1999), Heidelberg (1999-2007) I had been coordinating several projects on numerical modelling of variable density groundwater flow and contaminant transport. 

Level set methods in applications like

• free groundwater table

• fire front propagation

• two-phase flows

• segmentation and object detection in biological images

• boundary of atherosclerotic lesion

• see a recent lecture on several related topics - download the PDF file with movies.

A selection of my papers on level set methods (downloadable):

Peter Frolkovic and Karol Mikula: Flux-based level set method: a finite volume method for evolving interfaces, Applied Numerical Mathematics, 4 (57), p. 436-454, 2007

Peter Frolkovic and Karol Mikula: High-resolution flux-based level set method, SIAM J. Sci. Comp. 29 (2), p. 579-597, 2007

P. Frolkovič: Application of level set method for groundwater flow with moving boundary. Advances in Water Resources, Volume 47, October 2012, Pages 56–66

P. Frolkovič, K. Mikula, J. Urbán: "Distance function and extension in normal direction for implicitly defined interfaces", Discrete and Continuous Dynamical Systems - Series S, 8 (5), p. 871-880

P. Frolkovič, K. Mikula and J. Urbán: Semi-implicit finite volume level set method for advective motion of interfaces in normal direction. Applied Numerical Mathematics, 95, pp. 214-228, 2015

P. Frolkovič, D. Logashenko, C. Wehner: Flux-based level-set method for two-phase flows on unstructured grids. Comput. Vis. Sci., 18, p.31-52, 2016

P. Frolkovic, K. Mikula: Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids. Applied Mathematics and Computation, Volume 329, Pages 129-142, 2018. 

Ray, N., Oberlander, J. & Frolkovic, P. Numerical investigation of a fully coupled micro-macro model for mineral dissolution and precipitation. Comput Geosci 23, 1173–1192 (2019). https://doi.org/10.1007/s10596-019-09876-x

Numerical modelling of groundwater flow and contaminant transport:

• • • advection dominated reactive coupled transport equations with different retardation factorsflux-based method of characteristics

    • precise numerical solution for nonlinear sorption isotherms 

• • download a related lecture here

  • density driven flows

    • consistent velocity approximation

    • correct solution of Elder problem

  • ...

A selection of my papers:

Peter Frolkovic, Peter Knabner: Consistent Velocity Approximations in Finite Element or Volume Discretizations of Density Driven Flow, In: "Computational Methods in WaterResources XI", Vol. 1 (A.A. Aldama et al., eds.), Computational Mechanics Publication, Southhampten, 1996, p. 93-100

Peter Frolkovic: Consistent velocity approximation for density driven flow and transport, In: Advanced Computational Methods in Engineering, Part 2: Contributed papers; (R. Van Keer at al., eds.), Shaker Publishing, Maastricht, 1998, p. 603-611

Peter Frolkovic, Hennie De Schepper: Numerical modelling of convection dominated transport coupled with density driven flow in porous media, Advances in Water Resources, 24 (1), 2001, p. 63-72

Peter Frolkovic: Flux-based methods of characteristics for coupled transport equations in porous media, Computing and Visualization in Science; 6 (2004), 173-184

Peter Frolkovic and Jozef Kacur: Semi-analytical solutions of contaminant transport equation with nonlinear sorption in 1D, Computational Geosciences, 3 (10), 279-290, 2006

P. Frolkovič, M. Lampe, G. Wittum: Numerical simulation of contaminant transport in groundwater using software tools of r^3t. Comput. Vis. Sci., 18, p.17-29.