arXiv subjects

Dirk Praetorius

Publications and source records attributed to Dirk Praetorius.

At least 73 records · Page 4Linked to original sources

arXiv2019

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

math.NA↗

arXiv2019

Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

For the discretization of the integral fractional Laplacian $(-Δ)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.

math.NA↗

arXiv2019

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

math.NA↗

arXiv2019

Adaptive isogeometric boundary element methods with local smoothness control

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

math.NA↗

arXiv2019

Optimal convergence behavior of adaptive FEM driven by simple (h-h/2)-type error estimators

For some Poisson-type model problem, we prove that adaptive FEM driven by the (h-h/2)-type error estimators from [Ferraz-Leite, Ortner, Praetorius, Numer. Math. 116 (2010)] leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.

math.NA↗

arXiv2019

Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics

We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes.

math.NA↗

arXiv2018

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.

physics.comp-ph↗

arXiv2018

Adaptive BEM with optimal convergence rates for the Helmholtz equation

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.

math.NA↗

arXiv2018

Convergence of adaptive stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.

math.NA↗

arXiv2018

Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs

We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).

math.NA↗

arXiv2018

Iterative solution and preconditioning for the tangent plane scheme in computational micromagnetics

The tangent plane scheme is a time-marching scheme for the numerical solution of the nonlinear parabolic Landau-Lifshitz-Gilbert equation (LLG), which describes the time evolution of ferromagnetic configurations. Exploiting the geometric structure of LLG, the tangent plane scheme requires only the solution of one linear variational form per time-step, which is posed in the discrete tangent space determined by the nodal values of the current magnetization. We develop an effective solution strategy for the arising constrained linear systems, which is based on appropriate Householder reflections. We derive possible preconditioners, which are (essentially) independent of the time-step, and prove that the preconditioned GMRES algorithm leads to linear convergence. Numerical experiments underpin the theoretical findings.

math.NA↗

arXiv2018

Optimal Adaptivity for the SUPG Finite Element Method

For convection dominated problems, the streamline upwind Petrov--Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution. Based on robust a posteriori error estimators, we propose an adaptive mesh-refining algorithm for SUPG and prove that the generated SUPG solutions converge at asymptotically optimal rates towards the exact solution.

math.NA↗

arXiv2018

Adaptive BEM with inexact PCG solver yields almost optimal computational costs

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space $\widetilde{H}^{-1/2}(Γ)$. The main results also hold for the hyper-singular integral equation with energy space $H^{1/2}(Γ)$.

math.NA↗

arXiv2017

Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation

Combining ideas from [Alouges et al. (Numer. Math., 128, 2014)] and [Praetorius et al. (Comput. Math. Appl., 2017)], we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau-Lifshitz-Gilbert (LLG) equation which is unconditionally convergent, formally (almost) second-order in time, and requires only the solution of one linear system per time-step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then, we extend the scheme to the coupled system of the Landau-Lifshitz-Gilbert equation with the eddy current approximation of Maxwell equations (ELLG). Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires only the solution of two linear systems per time-step.

math.NA↗

arXiv2017

Convergence of an implicit-explicit midpoint scheme for computational micromagnetics

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau-Lifschitz-Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006) (Convergence of an implicit finite element method for the Landau-Lifschitz-Gilbert equation. SIAM J. Numer. Anal. 44), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams-Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.

math.NA↗

arXiv2017

Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection.

math.NA↗

arXiv2017

Rate optimal adaptive FEM with inexact solver for nonlinear operators

We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.

math.NA↗

arXiv2017

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge2$. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (resp. the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

math.NA↗