On Thursday, July 16 Fritz Kretzschmar successfully defended his thesis entitled

The Discontinuous Galerkin Trefftz Method

Congratulations Fritz!

EDIT Feb/2/2016: Please find his thesis online here.

Reply

On Thursday, July 16 Fritz Kretzschmar successfully defended his thesis entitled

The Discontinuous Galerkin Trefftz Method

Congratulations Fritz!

EDIT Feb/2/2016: Please find his thesis online here.

Patrick Sanan, Dave May and I submitted a paper entitled

Pipelined, Flexible Krylov Subspace Methods

The abstract reads

pyBlaSch is an open-source Python code demonstrating option valuation via the solution of the Black-Scholes equation
## Implementation

## Getting the code

\(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} – rV = 0.\)

It is a parabolic partial differential equation involving the option price \(V,\) the price of the underlying stock \(S,\) the volatility \(\sigma,\) and the risk free rate \(r.\) As it is simple to account for annualized dividend payments \(d\), these are included in the code, too.

- Finite differences discretization of the spatial derivative operators
- Runge-Kutta schemes for the integration in time (currently first to 4th order low-storage schemes)
- Currently only simple Dirichlet type boundary conditions are implemented
- Types of options are European and Binary call/put
- Derived quantities: Greeks Delta and Gamma, Put-Call parity implied price

The code design follows an object-oriented programming paradigm and was done with extensibility in mind.

pyBlaSch is open-source software licensed under the MIT license and available on Bitbucket at https://bitbucket.org/saschaschnepp/pyblasch.

Fritz Kretzschmar, Herbert Egger, Farzad Ahmadi, Nabil Nowak, Vadim A. Markel, Igor Tsukerman and myself contributed a chapter entitled

The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization

to the book Finite Difference Methods,Theory and Applications Volume 9045 of the series Lecture Notes in Computer Science. The abstract reads

Fritz Kretzschmar, Andrea Moiola, Ilaria Perugia and I prepared a paper on the a priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems.

The abstract reads

We present and analyse a space-time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space-time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al., and of Monk and Richter. For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Please find the preprint on arXiv.

For more information on the DG Trefftz method please see the posts tagged Trefftz DG.

EDIT Oct/28/2015: The paper got accepted by IMA Journal of Numerical Analysis

EDIT Dec/18/2015: Please find the final paper online.

Fritz Kretzschmar, Herbert Egger, Thomas Weiland and I submitted a paper on a space-time discontinuous Galerkin Trefftz method for the time-dependent Maxwell’s equations. Trefftz methods require the basis functions to fulfill the underlying PDEs in an exact sense. Consequently, vectorial basis functions in a space-time setting have to be considered. It is shown that we obtain a largely reduced number of degrees of freedom.

EDIT: The paper was published in the SIAM Journal on Scientific Computing (SISC) 37(5).

The abstract reads

We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin method based on Trefftz polynomials. This method fits into an abstract framework for space-time discontinuous Galerkin methods for which we can prove consistency, stability, and energy dissipation without the need to completely specify the approximation spaces in detail. Any method of such a general form results in an implicit time-stepping scheme with some basic stability properties. For the local approximation on each space-time element, we then consider Trefftz polynomials, i.e., the subspace of polynomials that satisfy Maxwell’s equations exactly on the respective element. We present an explicit construction of a basis for the local Trefftz spaces in two and three dimensions and summarize some of their basic properties. Using local properties of the Trefftz polynomials, we can establish the well-posedness of the resulting discontinuous Galerkin Trefftz method. Consistency, stability, and energy dissipation then follow immediately from the results about the abstract framework. The method proposed in this paper therefore shares many of the advantages of more standard discontinuous Galerkin methods, while at the same time, it yields a substantial reduction in the number of degrees of freedom and the cost for assembling. These benefits and the spectral convergence of the scheme are demonstrated in numerical tests.

Please find the preprint on arXiv.

For more information on the DG Trefftz method please see also our previous paper, which includes the exact treatment of inhomogeneous materials in partially filled cells as immersed boundaries and this link regarding transparent boundary conditions.

**UPDATE** May, 28, 2015: The paper was accepted.

The slides of our talk (Dominic Charrier, Dave May and myself) presented at the European Geosciences Union General Assembly (EGU) 2015 are online.

A discontinuous Galerkin method for studying elasticity and variable viscosity Stokes problems