[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...
[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).
[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!
[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.
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.