Gamma-convergent projection-free finite element methods for nematic liquid crystals: The Ericksen model

The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable, and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projection-free and thus circumvents the use of weakly acute meshes, which are quite restrictive in 3D but are required by recent algorithms for convergence. We prove stability and $Gamma$-convergence properties of the new method in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that controls the violation of the unit-length constraint of the director. We present several simulations in 2D and 3D that document the performance of the proposed scheme and its ability to capture quite intriguing defects.

Numerical analysis of the Landau-Lifshitz-Gilbert equation with inertial effects

We consider the numerical approximation of the inertial Landau-Lifshitz-Gilbert (iLLG) equation, which describes the dynamics of the magnetization in ferromagnetic materials at subpicosecond time scales. We propose and analyze two fully discrete numerical schemes based on two different approaches: The first method is based on a reformulation of the problem as a linear constrained variational formulation for the time derivative of the magnetization. The second method exploits a reformulation of the problem as a first order system in time for the magnetization and the angular momentum. Both schemes are implicit, based on first-order finite elements, and the constructed numerical approximations satisfy the inherent unit-length constraint of iLLG at the vertices of the underlying mesh. For both schemes, we establish a discrete energy law and prove convergence of the approximations towards a weak solution of the problem. Numerical experiments validate the theoretical results and show the applicability of the methods for the simulation of ultrafast magnetic processes.

Quantum Monte Carlo determination of the principal Hugoniot of deuterium

We present Coupled Electron-Ion Monte Carlo results for the principal Hugoniot of deuterium together with an accurate study of the initial reference state of shock wave experiments. We discuss the influence of nuclear quantum effects, thermal electronic excitations, and the convergence of the energy potential surface by wave function optimization within Variational Monte Carlo and Projection Quantum Monte Carlo methods. Compared to a previous study, the new calculations also include low pressure-temperature (P,T) conditions resulting in close agreement with experimental data, while our revised results at higher (P,T) conditions still predict a more compressible Hugoniot than experimentally observed.

Two-level a posteriori error estimation for adaptive multilevel stochastic Galerkin FEM

The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side function depend on infinitely many (uncertain) parameters. We introduce a two-level a posteriori estimator to control the energy error in multilevel stochastic Galerkin approximations for this class of PDE problems. We prove that the two-level estimator always provides a lower bound for the unknown approximation error, while the upper bound is equivalent to a saturation assumption. We propose and empirically compare three adaptive algorithms, where the structure of the estimator is exploited to perform spatial refinement as well as parametric enrichment. The paper also discusses implementation aspects of computing multilevel stochastic Galerkin approximations.

The saturation assumption yields optimal convergence of two-level adaptive BEM

We consider the convergence of adaptive BEM for weakly-singular and hypersingular integral equations associated with the Laplacian and the Helmholtz operator in 2D and 3D. The local mesh-refinement is driven by some two-level error estimator. We show that the adaptive algorithm drives the underlying error estimates to zero. Moreover, we prove that the saturation assumption already implies linear convergence of the error with optimal algebraic rates.

Computational micromagnetics with Commics

We present our open-source Python module Commics for the study of the magnetization dynamics in ferromagnetic materials via micromagnetic simulations. It implements state-of-the-art unconditionally convergent finite element methods for the numerical integration of the Landau-Lifshitz-Gilbert equation. The implementation is based on the multiphysics finite element software Netgen/NGSolve. The simulation scripts are written in Python, which leads to very readable code and direct access to extensive post-processing. Together with documentation and example scripts, the code is freely available on GitLab.

Correlation energies of the high-density spin-polarized electron gas to meV accuracy

We present a novel combination of quantum Monte Carlo methods and a finite size extrapolation framework with which we calculate the thermodynamic limit of the exact correlation energy of the polarized electron gas at high densities to meV accuracy, $-40.44(5)$ and $-31.70(4)$ mHa at $r_{rm s}=0.5$ and $1$, respectively. The fixed-node error is characterized and found to exceed $1$ mHa, and we show that the magnitude of the correlation energy of the polarized electron gas is underestimated by up to $6$ meV by the Perdew-Wang parametrization, for which we suggest improvements.