[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!
[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.
[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
[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
Efficient p-stable Boundary Element Methods -
A Matlab/C++ Software Package for 2D-boundary element methods with a focus on p and hp BEM. Do you think your code can use high polynomial degrees?
Then check this out. It's impressive.
Hilbert Is a Lovely Boundary
Element Research Tool - A Matlab/C++ Software Package for
adaptive 2D-boundary element methods which was developed in the above research project.