# Michael Karkulik

Departamento de Matemática
Oficina 248
Avenida España 1680
Valparaíso, Chile

Email: michael.karkulik that special sign usm.cl News : Research : Teaching : Links

## News

[14.08.2018] Here is a new work in collaboration with Markus Melenk. The fractional Laplacian is a nonlocal operator, and if you discretize it with finite elements you end up with a dense system matrix. You can approximate this matrix by blockwise low-rank matrices (H-matrices) and the error behaves exponentially in the rank. This is easy to see because you can write the fractional Laplacian as an operator with an asymptotically smooth kernel. So far, so good. In our new work we show that the same is true for the inverse of the system matrix.
[15.05.2018] Together with Thomas Führer we analyzed time stepping methods for parabolic problems based on fist-order system least-squares finite elements. For general spatial elliptic operators, optimal error a priori error estimates were not known up to know. You can find the preprint here.
[26.04.2017] Here is a new manuscript in collaboration with Thomas Führer, Norbert Heuer, and Rodolfo Rodríguez. In this earlier work we coupled DPG on a bounded domain with boundary elements in the exterior. Now, we decompose a domain into two parts and use DPG on one part and classical finite elements on the other to solve a general second order elliptic equation.
[12.04.2017] I wrote a paper which deals with a time-fraction diffusion equation. This equation looks (and behaves, in some sense) like the famous heat equation, but instead of one time derivative it has a fractional time derivative of order between 1/2 and 1. I was interested in having a variational formulation of this equation which resembles the well-known variational formulation of the heat equation (see your favourite PDE textbook). To that end, I had to deal with Sobolev-Bochner spaces and vector-valued extensions of scalar-valued operators.
[01.08.2016] I have joined the Departamento de Matemática at Universidad Técnica Federico Santa María.
[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.