arXiv subjects

Christian Kreuzer

Publications and source records attributed to Christian Kreuzer.

At least 19 recordsLinked to original sources

arXiv2025

A~posteriori error analysis for optimization with PDE constraints

We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control.

math.NA↗

arXiv2025

Strictly equivalent a~posteriori error estimators for quasi-optimal nonconforming methods

We devise a posteriori error estimators for quasi-optimal nonconforming finite element methods approximating symmetric elliptic problems of second and fourth order. These estimators are defined for all source terms that are admissible to the underlying weak formulations. More importantly, they are equivalent to the error in a strict sense. In particular, their data oscillation part is bounded by the error and, furthermore, can be designed to be bounded by classical data oscillations. The estimators are computable, except for the data oscillation part. Since even the computation of some bound of the oscillation part is not possible in general, we advocate to handle it on a case-by-case basis. We illustrate the practical use of two estimators obtained for the Crouzeix-Raviart method applied to the Poisson problem with a source term that is not a function and its singular part with respect to the Lebesgues measure is not aligned with the mesh.

math.NA↗

arXiv2025

Pressure robust finite element discretizations of the nonlinear Stokes equations

We present first-order nonconforming Crouzeix-Raviart discretizations for the nonlinear generalized Stokes equations with $(r,ε)$-structure. Thereby the velocity-errors are independent of the pressure-error; i.e., the method is pressure robust. This improves suboptimal rates previously experienced for non pressure robust methods.

math.NA↗

arXiv2020

Convergence of adaptive discontinuous Galerkin methods (corrected version of [Math. Comp. 87 (2018), no. 314, 2611--2640])

We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element methods by Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707--737].

math.NA↗

arXiv2020

On the threshold condition for Dörfler marking

It is an open question if the threshold condition $θ< θ_\star$ for the Dörfler marking parameter is necessary to obtain optimal algebraic rates of adaptive finite element methods. We present a (non-PDE) example fitting into the common abstract convergence framework (axioms of adaptivity) and which is potentially converging with exponential rates. However, for Dörfler marking $θ> θ_\star$ the algebraic converges rate can be made arbitrarily small.

math.NA↗

arXiv2020

Quasi-optimal and pressure robust discretizations of the Stokes equations by moment- and divergence-preserving operators

We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasi-optimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the load involving a linear operator, which maps discontinuous Galerkin test functions onto conforming ones thereby preserving the discrete divergence and certain moment conditions on faces and elements.

math.NA↗

arXiv2019

Convergence of an adaptive $C^0$-interior penalty Galerkin method for the biharmonic problem

We develop a basic convergence analysis for an adaptive $\textsf{C}^0\textsf{IPG}$ method for the Biharmonic problem, which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoulis [Math. Comp. 87 (2018), no.~314, 2611--2640], here we have to deal with the fact that the Lagrange finite element spaces may possibly contain no proper $C^1$-conforming subspace. This prevents from a straight forward generalisation and requires the development of some new key technical tools.

math.NA↗

arXiv2019

Quasi-best approximation in optimization with PDE constraints

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regularization parameter. Furthermore, if the operators of control-action and observation are compact, this quasi-best-approximation constant becomes independent of the Tikhonov parameter as the meshsize tends to $0$ and we give quantitative relationships between meshsize and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

math.NA↗

arXiv2019

Oscillation in a posteriori error estimation

In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite information. The new twist in our approach is a locally stable projection onto discretized residuals.

math.NA↗

arXiv2019

Quasi-optimal and pressure robust discretizations of the Stokes equations by new augmented Lagrangian formulations

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure robust, in the sense that the velocity $H^1$-error is proportional to the best $H^1$-error to the analytical velocity. This shows that such a property can be achieved without using conforming and divergence-free pairs. We bound also the pressure $L^2$-error, only in terms of the best approximation errors to the analytical velocity and the analytical pressure. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by Discontinuous Galerkin methods.

math.NA↗

arXiv2018

Temporally Resolved Intensity Contouring (TRIC) for characterization of the absolute spatio-temporal intensity distribution of a relativistic, femtosecond laser pulse

Today's high-power laser systems are capable of reaching photon intensities up to $10^{22}$ W/cm^2, generating plasmas when interacting with material. The high intensity and ultrashort laser pulse duration (fs) make direct observation of plasma dynamics a challenging task. In the field of laser-plasma physics and especially for the acceleration of ions, the spatio-temporal intensity distribution is one of the most critical aspects. We describe a novel method based on a single-shot (i.e. single laser pulse) chirped probing scheme, taking nine sequential frames at framerates up to THz. This technique, to which we refer as temporally resolved intensity contouring (TRIC) enables single-shot measurement of laser-plasma dynamics. Using TRIC, we demonstrate the reconstruction of the complete spatio-temporal intensity distribution of a high-power laser pulse in the focal plane at full pulse energy with sub picosecond resolution.

physics.plasm-ph↗

arXiv2018

I-BEAT: New ultrasonic method for single bunch measurement of ion energy distribution

The shape of a wave carries all information about the spatial and temporal structure of its source, given that the medium and its properties are known. Most modern imaging methods seek to utilize this nature of waves originating from Huygens' principle. We discuss the retrieval of the complete kinetic energy distribution from the acoustic trace that is recorded when a short ion bunch deposits its energy in water. This novel method, which we refer to as Ion-Bunch Energy Acoustic Tracing (I-BEAT), is a generalization of the ionoacoustic approach. Featuring compactness, simple operation, indestructibility and high dynamic ranges in energy and intensity, I-BEAT is a promising approach to meet the needs of petawatt-class laser-based ion accelerators. With its capability of completely monitoring a single, focused proton bunch with prompt readout it, is expected to have particular impact for experiments and applications using ultrashort ion bunches in high flux regimes. We demonstrate its functionality using it with two laser-driven ion sources for quantitative determination of the kinetic energy distribution of single, focused proton bunches.

physics.plasm-ph↗

arXiv2017

Adaptive Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

We develop the a posteriori error analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi--valued, maximal monotone $r$-graph, with $\frac{2d}{d+1}<r<\infty$. We establish upper and lower bounds on the finite element residual, as well as the local stability of the error bound. We then consider an adaptive finite element approximation of the problem, and, under suitable assumptions, we show the weak convergence of the adaptive algorithm to a weak solution of the boundary-value problem. The argument is based on a variety of weak compactness techniques, including Chacon's biting lemma and a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions, introduced by L. Diening, C. Kreuzer and E. Süli [Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal., 51(2), 984--1015].

math.NA↗

arXiv2016

A convergent time-space adaptive dG(s) finite element method for parabolic problems motivated by equal error distribution

We shall develop a fully discrete space-time adaptive method for linear parabolic problems based on new reliable and efficient a posteriori analysis for higher order dG(s) finite element discretisations. The adaptive strategy is motivated by the principle of equally distributing the a posteriori indicators in time and the convergence of the method is guaranteed by the uniform energy estimate from [KreuzerMöllerSchmidtSiebert:12] under minimal assumptions on the regularity of the data.

math.NA↗

arXiv2015

Efficient quasi-monoenergetic ion beams up to 18 MeV/nucleon via self-generated plasma fields in relativistic laser plasmas

Table-top laser-plasma ion accelerators seldom achieve narrow energy spreads, and never without serious compromises in efficiency, particle yield, etc. Using massive computer simulations, we identify a self-organizing scheme that exploits persisting self-generated plasma electric (~TV/m) and magnetic (~10${}^{4}$ Tesla) fields to reduce the ion energy spread after the laser exits the plasma - separating the ion acceleration from the energy spread reduction. Consistent with the scheme, we experimentally demonstrate aluminum and carbon ion beams with narrow spectral peaks at energies up to 310 MeV (11.5 MeV/nucleon) and 220 MeV (18.3 MeV/nucleon), respectively, with high conversion efficiency (~5%, i.e., 4J out of 80J laser). This is achieved with 0.12 PW high-contrast Gaussian laser pulses irradiating planar foils with optimal thicknesses of up to 250 nm that scale with laser intensity. When increasing the focused laser intensity fourfold (by reducing the focusing optic f/number twofold), the spectral-peak energy increases twofold. These results pave the way for next generation compact accelerators suitable for applications. For example, 400 MeV (33.3 MeV/nucleon) carbon-ion beam with narrow energy spread required for ion fast ignition could be generated using PW-class lasers.

physics.plasm-ph↗

arXiv2015

Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems

We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming approximations of the Poisson problem to nonconforming Crouzeix-Raviart approximations of the Poisson and the Stokes problem in 2D. As a consequence, we obtain instance optimality of an AFEM with a modified maximum marking strategy.

math.NA↗

arXiv2014

Instance optimality of the adaptive maximum strategy

In this paper, we prove that the standard adaptive finite element method with a (modified) `maximum marking strategy' is `instance optimal' for the `total error', being the sum of the energy error and the oscillation. This result will be derived in the model setting of Poisson's equation on a polygon, linear finite elements, and conforming triangulations created by newest vertex bisection.

math.NA↗

arXiv2013

Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi--valued, maximal monotone $r$-graph, with $1<r<\infty$. Using a variety of weak compactness techniques, including Chacon's biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter $h$ tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions.

math.NA↗