arXiv subjects

Nils Margenberg

Publications and source records attributed to Nils Margenberg.

At least 19 recordsLinked to original sources

arXiv2026

Goal-oriented space-time adaptivity for the Navier--Stokes equations based on the dual weighted residual method

This work presents a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method together with space-time mesh adaptivity for the Navier--Stokes equations. The resulting nonlinear algebraic systems on the space-time slabs are solved by Newton's method with GMRES, preconditioned by a slab-wise geometric multigrid method. This combination yields reliable control of target quantities on computationally feasible space-time meshes together with a robust and efficient solution of the algebraic systems. The implementation is based on a MPI-parallel programming model in the deal.II library. Further ingredients are a discontinuous Galerkin discretization in time and inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes. The performance of the approach is investigated in benchmark computations with regard to accuracy, efficiency, and stability.

math.NA↗

arXiv2026

Stochastic Galerkin and Monte-Carlo methods for parabolic problems: Numerical performance of variational matrix-free approximations

Stochastic Galerkin methods offer unexplored potential for the numerical simulation of parabolic problems with random variables, in particular if they are combined with variational discretizations of the space and time variables. Due to the high dimensionality, the solution of the arising algebraic systems do not become feasible without efficient solvers, preconditioners, and software architectures. A stochastic Galerkin discretization with an embedded slabwise finite element approximation of the space and time variables is proposed and analyzed numerically. For solving the linear systems, GMRES iterations are block-preconditioned by a geometric multigrid (GMG) technique using a local Vanka smoother for the space-time subsystems. Monte-Carlo methods are also used for solving random parabolic problems and studied here for the purpose of comparison. The Monte-Carlo approach is built on the space-time finite element formulation together with the GMRES-GMG solver technology. All algorithms have been implemented in a unified matrix-free framework based on the deal.II software library. Comparative numerical evaluations illustrate the performance properties of both approaches, including convergence of the discretizations and statistics of the algebraic solver. Superiority of the stochastic Galerkin approach is observed.

math.NA↗

arXiv2026

Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II

In this article, we solve the Stokes and Navier-Stokes equations with the deal$.$II finite-element library. In particular, we use its multigrid, adaptive-mesh, and matrix-free infrastructures to design efficient linear and nonlinear iterative solvers, respectively. We solve the stationary Stokes equations on hp-adaptive meshes with a hp-multigrid approach, the transient Stokes equations with space-time finite elements and space-time multigrid, and, finally, the stabilized incompressible Navier-Stokes equations on locally refined meshes with a monolithic multigrid solver. The selected examples underline the flexibility and modularity of the multigrid infrastructure of deal$.$II.

math.NA↗

arXiv2026

A Scalable Monolithic Modified Newton Multigrid Framework for Time-Dependent $p$-Navier-Stokes Flow

Fully implicit tensor-product space-time discretizations of time-dependent $(p,\delta)$-Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime $1 0$ and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

math.NA↗

arXiv2026

A Monolithic hp Space-Time Multigrid Preconditioned Newton-Krylov Solver for Space-Time FEM applied to the Incompressible Navier-Stokes Equations

We present a monolithic hp space-time multigrid method (hp-STMG) for tensor-product space-time finite element discretizations of the incompressible Navier-Stokes equations. We employ mapped inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\mathrm{disc}}$ in space and a slabwise discontinuous Galerkin DG($k$) discretization in time. The resulting fully coupled nonlinear systems are solved by Newton-GMRES preconditioned with hp-STMG, combining geometric coarsening in space with polynomial coarsening in space and time. Our main contribution is an hp-robust and practically efficient extension of space-time multigrid to Navier-Stokes: matrix-free operator evaluation is retained via column-wise, state-dependent spatial kernels; the nonlinear convective term is handled by a reduced, order-preserving time quadrature. Robustness is ensured by an inexact space-time Vanka smoother based on patch models with single time point evaluation. The method is implemented in the matrix-free multigrid framework of deal.II and demonstrates h- and p-robust convergence with robust solver performance across a range of Reynolds numbers, as well as high throughput in large-scale MPI-parallel experiments.

math.NA↗

arXiv2026

Anisotropic hp space-time adaptivity and goal-oriented error control for convection-dominated problems

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-dominated problems. Using elementwise p-anisotropic finite element spaces, the estimator is elementwise separated with respect to the single directions in space and time. This naturally leads to adaptive, anisotropic hp-refinement (h-anisotropic refinement and elementwise anisotropic p-enrichment). We employ discontinuous elements in space and time, which are well suited for problems with high Peclet numbers. Efficiency and robustness of the underlying algorithm are demonstrated for different goal functionals. The directional error indicators quantify anisotropy of the solution with respect to the goal, and produce hp-refinements that efficiently capture sharp layers. Numerical examples in up to three spatial dimensions demonstrate the superior performance of the proposed method compared to isotropic h and hp adaptive refinement using established benchmarks for convection-dominated transport.

math.NA↗

arXiv2026

A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries

The deep neural network multigrid solver (DNN-MG) combines a coarse-grid finite element simulation with a deep neural network that corrects the solution on finer grid levels, thereby improving the computational efficiency. In this work, we discuss various design choices for the DNN-MG method and demonstrate significant improvements in accuracy and generalizability when applied to the solution of the nonstationary Navier-Stokes equations. We investigate the stability of the hybrid simulation and show how the neural networks can be made more robust with the help of replay buffers. After an initial single-step training, we run the hybrid simulation for extended periods and compute new reference solutions of the neural network perturbed state for each step. By retraining on this data, the error caused by the neural network over multiple time-steps due to distributional shift can be effectively reduced without the need for a differentiable numerical solver. Furthermore, we compare multiple neural network architectures, including recurrent neural networks and Transformers, and study their ability to utilize more information from an increased temporal and spatial receptive field. Transformers allow us to make use of information from cells outside the predicted patch even with unstructured meshes while maintaining the locality of our approach. This can further improve the accuracy of DNN-MG without a significant impact on performance.

math.NA↗

arXiv2025

A Hybrid Neural Network-Finite Element Method for the Viscous-Plastic Sea-Ice Model

We present an efficient hybrid Neural Network-Finite Element Method (NN-FEM) for solving the viscous-plastic (VP) sea-ice model. The VP model is widely used in climate simulations to represent large-scale sea-ice dynamics. However, the strong nonlinearity introduced by the material law makes VP solvers computationally expensive, with the cost per degree of freedom increasing rapidly under mesh refinement. High spatial resolution is particularly required to capture narrow deformation bands known as linear kinematic features in viscous-plastic models. To improve computational efficiency in simulating such fine-scale deformation features, we propose to enrich coarse-mesh finite element approximations with fine-scale corrections predicted by neural networks trained with high-resolution simulations. The neural network operates locally on small patches of grid elements, which is efficient due to its relatively small size and parallel applicability across grid patches. An advantage of this local approach is that it generalizes well to different right-hand sides and computational domains, since the network operates on small subregions rather than learning details tied to a specific choice of boundary conditions, forcing, or geometry. The numerical examples quantify the runtime and evaluate the error for this hybrid approach with respect to the simulation of sea-ice deformations. Applying the learned network correction enables coarser-grid simulations to achieve qualitatively similar accuracy at approximately 11 times lower computational cost relative to the high-resolution reference simulations. Moreover, the learned correction accelerates the Newton solver by up to 10% compared to runs without the correction at the same mesh resolution.

math.NA↗

arXiv2025

Multi-goal-oriented anisotropic error control and mesh adaptivity for time-dependent convection-dominated problems

In this work, we present an anisotropic multi-goal error control based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. This multi-goal oriented approach allows for an accurate and efficient error control with regard to several quantities of interest simultaneously. Using anisotropic interpolation and restriction operators, we obtain elementwise error indicators in space and time, where the spatial indicators are additionally separated with respect to the single directions. The directional error indicators quantify anisotropy of the solution with respect to the goals, and produce adaptive, anisotropic meshes that efficiently capture layers. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high P\'{e}clet numbers. Numerical examples show efficiency and robustness of the proposed approach for several goal quantities using established benchmarks for convection-dominated transport.

math.NA↗

arXiv2025

An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations

We present a monolithic $hp$ space-time multigrid method for tensor-product space-time finite element discretizations of the Stokes equations. Geometric and polynomial coarsening of the space-time mesh is performed, and the entire algorithm is expressed through rigorous mathematical mappings. For the discretization, we use inf-sup stable pairs $\mathbb Q_{r+1}/\mathbb P_{r}^{\text{disc}}$ of elements in space and a discontinuous Galerkin (DG$(k)$) discretization in time with piecewise polynomials of order $k$. The key novelty of this work is the application of $hp$ multigrid techniques in space and time, facilitated and accelerated by the matrix-free capabilities of the deal$.$II library. While multigrid methods are well-established for stationary problems, their application in space-time formulations encounter unique challenges, particularly in constructing suitable smoothers. To overcome these challenges, we employ space-time cell and vertex star patch based Vanka smoothers. Extensive tests on high-performance computing platforms demonstrate the efficiency of our $ hp $ multigrid approach on problem sizes exceeding a trillion degrees of freedom (dofs), sustaining throughputs of hundreds of millions of dofs per second.

math.NA↗

arXiv2024

Well-posedness and exponential stability of dispersive nonlinear Maxwell equations with PML: An evolutionary approach

This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models that include materials with complex dielectric properties, dispersion, and discontinuities. We use a Hilbert space framework to address these complex physical models in nonlinear optics. While our focus is on the first-order formulation in space and time, higher solution regularity recovers and equates to the second-order formulation. We incorporate perfectly matched layers (PMLs), which model absorbing boundary conditions, to facilitate the development of numerical methods. We demonstrate that the combined system remains well-posed and exponentially stable. Our approach applies to a broad class of partial differential equations (PDEs) and accommodates materials with nonlocal behavior in space and time. The contribution of this work is a unified framework for analyzing wave interactions in advanced optical materials.

math.AP↗

arXiv2024

A Space-Time Multigrid Method for Space-Time Finite Element Discretizations of Parabolic and Hyperbolic PDEs

We present a space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order continuous and discontinuous variational time discretizations with spatial finite element discretizations. The effectiveness of multigrid methods in large-scale stationary problems is well established. However, their application in the space-time context poses significant challenges, mainly due to the construction of suitable smoothers. To address these challenges, we develop a space-time cell-wise additive Schwarz smoother and demonstrate its effectiveness on the heat and acoustic wave equations. The matrix-free framework of the {\ttfamily deal.II} library supports various multigrid strategies, including $h$-, $p$-, and $hp$-refinement across spatial and temporal dimensions. Extensive empirical evidence, provided through scaling and convergence tests on high-performance computing platforms, demonstrate high performance on perturbed meshes and problems with heterogeneous and discontinuous coefficients. Throughputs of over a billion degrees of freedom per second are achieved on problems with more than a trillion global degrees of freedom. The results prove that the space-time multigrid method can effectively solve complex problems in high-fidelity simulations and show great potential for use in coupled problems.

math.NA↗

arXiv2024

The Biot-Allard poro-elasticity system: equivalent forms and well-posedness

We consider the fully dynamic Biot-Allard model, which includes memory effects. Convolution integrals in time model the history of the porous medium. We use a series representation of the dynamic permeability in the frequency domain to rewrite the equations in a coupled system without convolution integrals, suitable for the design of efficient numerical approximation schemes. The main result is the well-posedness of the system, proved by the abstract theory of R. Picard for evolutionary problems.

math.AP↗

arXiv2023

DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations

We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment a finite element simulation on coarse grids with fine scale information obtained using deep neural networks. The neural network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent generalizability of the method. Since the network only processes data of small sub-areas, it never ``sees'' the global problem and thus does not learn false biases. We describe the method with a focus on the interplay between the finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracy with a coarse grid and thus reduce the computation time by orders of magnitude.

math.NA↗

arXiv2023

Optimal Dirichlet Boundary Control by Fourier Neural Operators Applied to Nonlinear Optics

We present an approach for solving optimal Dirichlet boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods in combination with collocation techniques. Thereby, $C^{l}$-regularity in time of the global discrete is ensured. The resulting simulation data is used to train solution operators that effectively leverage the higher regularity of the training data. The solution operator is represented by Fourier Neural Operators and Gated Recurrent Units and can be used as the forward solver in the optimal Dirichlet boundary control problem. The proposed algorithm is implemented and tested on modern high-performance computing platforms, with a focus on efficiency and scalability. The effectiveness of the approach is demonstrated on the problem of generating Terahertz radiation in periodically poled Lithium Niobate, where the neural network is used as the solver in the optimal control setting to optimize the parametrization of the optical input pulse and maximize the yield of $0.3\,$THz-frequency radiation. We exploit the periodic layering of the crystal to design the neural networks. The networks are trained to learn the propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. Our results indicate that the proposed method can achieve a significant speedup in computation time compared to classical methods. A comparison of our results to experimental data shows the potential to revolutionize the way we approach optimization problems in nonlinear optics.

math.NA↗

arXiv2023

Benchmark computations of dynamic poroelasticity

We present benchmark computations of dynamic poroelasticity modeling fluid flow in deformable porous media by a coupled hyperbolic-parabolic system of partial differential equations. A challenging benchmark setting and goal quantities of physical interest for this problem are proposed. Computations performed by space-time finite element approximations with continuous and discontinuous discretizations of the time variable are summarized. By this work we intend to stimulate comparative studies by other research groups for the evaluation of dynamic poroelasticity solver regarding the accuracy of discretization techniques, the efficiency and robustness of iterative methods for the linear systems and the arrangement of the model equations in terms of their variables (two-field or multi-field formulations).

math.NA↗

arXiv2023

An energy-efficient GMRES-Multigrid solver for space-time finite element computation of dynamic poro- and thermoelasticity

We present families of space-time finite element methods (STFEMs) for a coupled hyperbolic-parabolic system of poro- or thermoelasticity. Well-posedness of the discrete problems is proved. Higher order approximations inheriting most of the rich structure of solutions to the continuous problem on computationally feasible grids are naturally embedded. However, the block structure and solution of the algebraic systems become increasingly complex for these members of the families. We present and analyze a robust geometric multigrid (GMG) preconditioner for GMRES iterations. The GMG method uses a local Vanka-type smoother. Its action is defined in an exact mathematical way. Due to nonlocal coupling mechanisms of unknowns, the smoother is applied on patches of elements. This ensures the damping of error frequencies. In a sequence of numerical experiments, including a challenging three-dimensional benchmark of practical interest, the efficiency of the solver for STFEMs is illustrated and confirmed. Its parallel scalability is analyzed. Beyond this study of classical performance engineering, the solver's energy efficiency is investigated as an additional and emerging dimension in the design and tuning of algorithms and their implementation on the hardware.

math.NA↗

arXiv2022

Accurate simulation of THz generation with Finite-Element Time Domain methods

We investigate the accurate full broadband simulation of complex nonlinear optical processes. A mathematical model and numerical simulation techniques in the time domain are developed to simulate complex nonlinear optical processes without the usual used slowly varying envelope approximation. We illustrate the accuracy by numerical simulations. Furthermore, they are used to elucidate THz generation in periodically poled Lithium Niobate (PPLN) including optical harmonic generation.

physics.optics↗