# Michael Karkulik

Fariborz Maseeh Department of Mathematics and Statistics
Portland State University
PO Box 751
Portland, Oregon 97207
Phone: +1 503 725 8289
Email: mkarkulik that special sign pdx.edu

Contact : News : Publications : Theses : Talks : Links

## News

[18.11.2015] I have joined the Department of Mathematics and Statistics at Portland State University.
[28.09.2015] Here is a new work in collaboration with Norbert Heuer. We present a DPG finite element method for reaction dominated diffusion problems. The standard energy norm for this kind of problems is weak for small diffusion coefficients, hence we use a special variational formulation which induces a stronger norm. Our theoretical results are robust for arbitrary diffusion parameter, and the numerical examples are very promising.
[10.08.2015] Take a look at our new manuscript. We show how to couple DPG finite elements with boundary elements, but we only compute optimal test functions for the finite element part. This way, the trial-to-test operator can still be computed locally, which is an improvement of this earlier work.
[03.08.2015] Here is a new work our group here at PUC did in collaboration with Vince Ervin. It's on a DPG strategy for a fractional differential equation. What is a fractional differential equation, you ask? An example for a fractional differential operator would be the square root of the derivative. Apply it twice to a function, then you get the usual derivative. Nice, isn't it?
[28.05.2015] The webpage of WONAPDE 2016 is online. If you have never been there: highly recommended!
[24.04.2015] Together with my former advisors and colleagues from Vienna, we extended one of our earlier preprints on local inverse estimates for boundary integral operators. The interesting point is that such operators are non-local, but not too much, and we still have local inverse estimates. The new version covers curved boundaries and the estimates are explicit in the polynomial degree.
[16.12.2014] Here is a new paper in collaboration with Norbert Heuer. It's about a DPG method for a finite element/boundary element coupling to solve a transmission problem.
[20.11.2014] I want to advertise two papers: The first one, in collaboration with Norbert Heuer, is about the DPG method with optimal test functions applied to integral equations. This allows us to solve a hypersingular integral equation with a completely discontinuous trial space. Here are some pictures! The second one, in collaboration with Markus Melenk, is about quasi-interpolation in the hp-context. What we do is basically mollifying with a spatially varying ε to turn a rough function into a smooth one, and then interpolate this smooth function. Another very nice application of this smoothing procedure is that we obtain local residual error estimates for hp-boundary elements.
[03.07.2014] Check out the revised version of our paper on H1 stability of the L2 projection.
[25.02.2014] Meanwhile in the publishing community...
[24.02.2014] The variational crime of approximating traces in the DPG method by discontinuous functions is analyzed in this new preprint with Norbert Heuer and Francisco Javier Sayas.
[06.02.2014] Together with Norbert Heuer and the ABEM Group from Vienna, I wrote an article that gives an overview of adaptive boundary element methods. It covers in particular the latest convergence and optimality results.
[16.12.2013] Monumental fail: the new Austrian government is going to close the Minstry of Science and Research and fold it into the Ministry of Economy. Read more (in German).
[03.12.2013] Together with Norbert Heuer, I wrote an article about an adaptive non-conforming boundary element method using Crouzeix-Raviart elements. Well, happy 40th birthday, Crouzeix-Raviart element!
[25.10.2013] On November 2, 2013, I will start with the FONDECYT project 3140614 "Efficient adaptive strategies for nonconforming boundary element methods", with supervision and help of Prof. Norbert Heuer. I will put here some funky pictures from experiments in the near future, come back some time if you are interested!
[08.10.2013] The board of the Austrian Science Fund wrote an open letter to the Austrian government (on the front page, in German).
[04.10.2013] The second part of the convergence analysis for adaptive BEM with data approximation can be found in this new preprint. While the first part was concerend with weakly-singular operators, the second part deals with the hypersingular operator. My favourite operator, by the way.
[29.08.2013] New Preprint! It extends our previous work on convergence rates for adaptive boundary elements. We show how to include data-approximation into the adaptive loop in an optimal way. Another nice feature is that we do not need any lower bound (efficiency) for the estimator, as we define the approximation classes in a different way as it was done until now.
[07.08.2013] Together with my former colleagues from the ABEM Group in Vienna, I wrote paper on a-posteriori error estimation for hypersingular integral equations. Besides the main topic, the paper contains some really nice results (Scott-Zhang with boundary conditions in a space without trace theorem, inverse estimates in fractional order Sobolev spaces...) Check it out!
[23.07.2013] My boss is a star on youtube!
[04.07.2013] Together with C.M. Pfeiler and D. Praetorius, I wrote an article about the H1 stability of L2 orthogonal projections onto lowest order finite elements in any space dimension. It generalizes our previous work. More to follow!
Here is the preprint on ZZ-type a posteriori error estimates in BEM.
My dissertation has been awarded the Dr. Körper-Prize by the Gesellschaft für angewandte Mathematik und Mechanik. Thank you!
The Gesellschaft für angewandte Mathematik und Mechanik also nominated my dissertation for the ECCOMAS PhD-Award 2012. Thank you!
The article "Quasi-optimal convergence rate of an adaptive boundary element method" is finally published.
In the April 2013 issue of the Notices of the American Mathematical Society, David A. Edwards wrote an interesting article about patents in mathematics. Here it is. The job adverdisment is clearly a coincidence.

## Publications

in Journals, Preprints and Reports, Proceedings

#### In Journals

[18] DPG method with optimal test functions for a transmission problem
with N. Heuer
accepted for publication in Comput. Math. Appl.
arXiv.1411.4753
[17] Local high-order regularization and applications to hp-methods
with J.M. Melenk
Comput. Math. Appl., 70(7) (2015), pp. 1606--1639
Elsevier, ASC Report 38/2014, arXiv.1411.5209
[16] Adaptive Crouzeix-Raviart boundary element method
with N. Heuer
ESAIM Math. Model. Numer. Anal., 49(4) (2015), pp. 1193--1217
EDPS, PUC MAT2014-006, arXiv:1312.0484
[15] Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, Part II: Hyper-singular integral equation
with M. Feischl, T. Führer, J.M. Melenk, and D. Praetorius
Electron. Trans. Numer. Anal., 44 (2015), pp. 153--176
RICAM, ASC Report 30/2013
[14] Stability of symmetric and nonsymmetric FEM-BEM couplings for nonlinear elasticity problems
with M. Feischl, T. Führer, and D. Praetorius
Numer. Math., 130(2) (2015), pp. 199--223
Springer, ASC Report 52/2012, arXiv:1212.2620
[13] Note on discontinuous trace approximation in the practical DPG method
with N. Heuer and F.-J. Sayas
Comput. Math. Appl., 68, (2014), pp. 1562--1568
Elsevier, PUC MAT2014-007, arXiv:1402.5169
[12] Adaptive boundary element methods: A posteriori error estimators, adaptivity, convergence, and implementation
M. Feischl, T. Führer, N. Heuer, M. Karkulik, D. Praetorius
Arch. Comput. Methods. Eng., 22(3) (2015), pp. 309--389
Springer, ASC Report 09/2014, PUC MAT2014-008, arXiv:1402.0744
[11] Energy norm based error estimators for adaptive BEM for hypersingular integral equations
M. Aurada, M. Feischl, T. Führer, M. Karkulik, D. Praetorius
Appl. Numer. Math., 95, (2015), pp. 15--35
Elsevier, ASC Report 22/2013
[10] ZZ-type a posteriori error estimators for adaptive boundary element methods on a curve
M. Feischl, T. Führer, M. Karkulik, and D. Praetorius
Eng. Anal. Bound. Elem., 38 (2014), pp. 49--60.
Elsevier, ASC Report 16/2013, arXiv:1306.5120
[9] Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, Part I: Weakly-singular integral equation
M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, D. Praetorius
Calcolo 51 (2014), pp. 485--508
Springer, ASC Report 24/2013
[8] HILBERT-A MATLAB Implementation of Adaptive 2D-BEM
M. Aurada, M. Ebner, M. Feischl, S. Ferraz-Leite, T. Führer, P. Goldenits, M. Karkulik, M. Mayr, and D. Praetorius
Numer. Algorithms 67(1) (2014), pp. 1--32
Springer, ASC Report 24/2011
[7] Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods
M. Aurada, M. Feischl, T. Führer, M. Karkulik, and D. Praetorius
Comput. Methods Appl. Math. 13(3) (2013), pp. 305--332
De Gruyter, ASC Report 15/2012
[6] On 2D newest vertex bisection: Optimality of mesh-closure and H^1-stability of L_2-projection
M. Karkulik, D. Pavlicek, and D. Praetorius
Constr. Approx. 38(2) (2013), pp. 213--234
Springer, ASC Report 10/2012, arXiv:1210.0367
[5] Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh-refinement
M. Karkulik, G. Of, D. Praetorius
Numer. Methods Partial Differential Equations 29(6) (2013), pp. 2081--2106
Wiley, ASC Report 20/2012
[4] Quasi-optimal convergence rate for an adaptive boundary element method
M. Feischl, M. Karkulik, J.M. Melenk, and D. Praetorius
SIAM. J. Numer. Anal. 51(2) (2013), pp. 1327--1348
SIAM, Article, ASC Report 28/2011,
[3] Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, and D. Praetorius
Comp. Mech., 51 (2013), pp. 399--419.
Springer, ASC Report 8/2012, arXiv:1211.4225
[2] A posteriori error estimates for the Johnson-Nédélec FEM-BEM coupling
M. Aurada, M. Feischl, M. Karkulik, and D. Praetorius
Eng. Anal. Bound. Elem., 36 (2012), pp. 255--266.
Elsevier, ASC Report 18/2011
[1] Convergence of adaptive BEM for some mixed boundary value problem
M. Aurada, S. Ferraz-Leite, P. Goldenits, M. Karkulik, M. Mayr, and D. Praetorius
Appl. Numer. Math., 62 (2012), pp. 226--245.
Elsevier, ASC Report 21/2010

#### Preprints and Technical Reports

[9] A robust DPG method for singularly perturbed reaction-diffusion problems
with N. Heuer
arXiv.1509.07560
[7] On the coupling of DPG and BEM
with T. Führer and N. Heuer
arXiv.1508.00630
[6] DPG method with optimal test functions for a fractional advection diffusion equation
with V.J. Ervin, T. Führer, and N. Heuer
arXiv.1507.06691
[5] Local inverse estimates for non-local boundary integral operators
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, and D. Praetorius
ASC Report 12/2015, arXiv.1504.04394
[4] Discontinuous Petrov-Galerkin boundary elements
with N. Heuer
arXiv.1408.5374
[3] L2-orthogonal projections onto lowest-order finite elements in Rd are H1-stable
M. Karkulik, C. M. Pfeiler, D. Praetorius
ASC Report 21/2013, arXiv:1307.0917
[2] Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM
M. Aurada, M. Feischl, T. Führer, M. Karkulik, J.M. Melenk, and D. Praetorius
ASC Report 07/2012
[1] HILBERT - a MATLAB implementation of adaptive BEM
M. Aurada, M. Ebner, S. Ferraz-Leite, P. Goldenits, M. Karkulik, M. Mayr, and D. Praetorius
ASC Report 44/2009

#### Proceedings

[7] FEM-BEM couplings without stabilization
M. Feischl, T. Führer, M. Karkulik, J. Melenk, D. Praetorius
Proceedings of IABEM 2013, (2013)
ASC Report 47/2012
M. Feischl, T. Führer, M. Karkulik, and D. Praetorius
Proceedings of IABEM 2013, (2013)
ASC Report 48/2012
[5] Novel inverse estimates for non-local operators
M. Feischl, T. Führer, M. Karkulik, J. Melenk, and D. Praetorius
Proceedings of IABEM 2013, (2013)
ASC Report 49/2012
[4] Residual a-posteriori error estimates in BEM: convergence of h-adaptive algorithms
M. Feischl, M. Karkulik, J. Melenk, and D. Praetorius
Proceedings of IABEM 2011, (2011)
ASC Report 21/2011
[3] Adaptive coupling of FEM and BEM: Simple error estimators and convergence
M. Aurada, M. Feischl, M. Karkulik, and D. Praetorius
Proceedings of IABEM 2011, (2011)
ASC Report 20/2011
[2] Adaptive coupling of FEM and BEM: Simple error estimators and convergence
M. Aurada, M. Feischl, M. Karkulik, and D. Praetorius
Proceedings in Applied Mathematics and Mechanics, PAMM, 11, (2011)
ASC Report 22/2011
[1] Adaptive coupling of FEM and BEM: Simple error estimators and convergence
M. Aurada, M. Feischl, M. Karkulik, and D. Praetorius
Proceedings of AfriCOMP 11, (2011)
ASC Report 35/2010

## Theses

Zur Konvergenz und Quasioptimalität adaptiver Randelementmethoden
PhD thesis, Vienna University of Technology, October 2012
pdf
Quasi-Interpolation Operators for hp-Finite Element Methods
Masters's thesis, Vienna University of Technology, 2009

## Talks

[-] A robust DPG method for singularly perturbed reaction-diffusion problems
WONAPDE 2016, Concepción, Chile, 11.01.2016 - 15.01.2016.
[-] Convergence of adaptive FE/BE coupling methods
ENUMATH 2015 - European Conference on Numerical Mathematics and Advanced Applications, Ankara, Turkey, 14.09.2015 - 18.09.2015.
[-] Convergence of adaptive FE/BE coupling methods
COMCA 2015 - XXIV Congreso de Matemática Capricornio, Iquique, Chile, 05.08.2015 - 07.08.2015.
[-] Nonsymmetric coupling of boundary elements and ultraweak finite elements and DPG method with optimal test functions
13th US National Congress on Computational Mechanics, San Diego, California, 26.07.2015 - 30.07.2015.
[-] DPG method with optimal test functions for a transmission problem
Caleta Numérica, Valparaíso, Chile, 05.12.2015.
[-] Nonsymmetric coupling of boundary elements and ultraweak finite elements and DPG method with optimal test functions
1.st Pan-American Congress on Computational Mechanics, Buenos Aires, Argentina, 27.04.2015 - 29.04.2015.
[-] Local high-order regularization and applications to hp-methods
La Serena Numérica II, La Serena, Chile, 14.01.2015 - 16.01.2015.
[-] Nonconforming DPG method
CMAM 2014 - International Conference on Computational Methods in Applied Mathematics, St. Wolfgang, Austria, 28.09.2014 - 04.10.2014.
[-] Nonconforming DPG method
COMCA 2014 - XXIII Congreso de Matemática Capricornio, Universidad de Atacama, Copiapó, Chile, 06.08.2014 - 08.08.2014.
[-] DPG boundary elements with optimal test functions on surfaces
Valparaíso Numérico IV, Valparaíso, Chile, 11.12.2013 - 13.12.2013.
Seminario de Análisis Numérico y Modelación Matemática, Universidad del Bío Bío, Concepción, 01.10.2013.
[-] Adaptive nonconforming boundary element methods
COMCA 2013 - XXII Congreso de Matemáticas, Universidad de La Serena, Chile, 31.07.2013 - 02.08.2013.
[-] Adaptive nonconforming boundary element methods
MAFELAP 2013, Brunel University, Uxbridge, England, 11.06.2013 - 14.06.2013.
WONAPDE 2013, Concepción, Chile, 14.01.2013 - 18.01.2013.
[-] Novel inverse estimates for non-local operators
IABEM 2013 Conference, Santiago, Chile, 09.01.2013-11.01.2013.
Valparaíso's Mathematics and its Applications Days (V-MAD III), Pontificia Universidad Católica de Valparaíso, Chile, 12.12.2012-14.12.2012.
[-] On the convergence and quasi-optimality of adaptive boundary element methods
Seminar of the Institute for Numerical Mathematics, Graz University of Technology, 07.11.2012.
[-] Novel inverse estimates for non-local operators
10th Workshop on Fast Boundary Element Methods in Industrial Applications, Hirschegg/Kleinwalsertal, 27.09.2012-30.09.2012.
[-] Quasi-optimal convergence rate for an adaptive boundary element method
Computational Methods in Applied Mathematics CMAM-5, Berlin, 30.07.2012-03.08.2012.
[-] A survey on adaptive boundary element methods
Fast BEM and BETI, Ostrava (Tschechien), 18.06.2012-19.06.2012.
[-] On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H1 Stability of L2-Projection
8th Austrian Numerical Analysis Day, Wien, 10.05.2012-11.05.2012
[-] Quasi-optimal convergence rate for an adaptive boundary element method
BEM on the Saar 2012, Universität des Saarlandes, 12.05.2012-16.05.2012.
[-] Residual a-posteriori error estimates in BEM: convergence of h-adaptive algorithms
IABEM 2011 Conference, Brescia, 05.09.2011-08.09.2011.
[-] hp-Quasi-Interpolation
Poster: Junior Scientist Conference, The City College of New York, 13.04.2011-15.04.2011.
[-] Application of Interpolation theory to adaptive 3D-BEM
8th Söllerhaus Workshop on Fast Boundary Element Methods in Industrial Applications, Hirschegg/Kleinwalsertal, 30.09.2010-03.10.2010.
[-] HILBERT - A Matlab Library for Adaptive 2D BEM
6th Austrian Numerical Analysis Day, Salzburg, 06.05.2010-07.05.2010
[-] Convergence of Data-Perturbed Adaptive Boundary Element Methods
WONAPDE 2010, Concepción (Chile), 11.01.2010-15.01.2010.