arXiv subjects

Jörn Wichmann

Publications and source records attributed to Jörn Wichmann.

12 recordsLinked to original sources

arXiv2025

$\textit{A Priori}$ Error Analysis for the $p$-Stokes Equations with Slip Boundary Conditions: A Discrete Leray Projection Framework

We present an $\textit{a priori}$ error analysis for the kinematic pressure in a fully-discrete finite-differences/-elements discretization of the unsteady $p$-Stokes equations, modelling non-Newtonian fluids. This system is subject to both impermeability and perfect Navier slip boundary conditions, which are incorporated either weakly via Lagrange multipliers or strongly in the discrete velocity space. A central aspect of the $\textit{a priori}$ error analysis is the discrete Leray projection, constructed to quantitatively approximate its continuous counterpart. The discrete Leray projection enables a Helmholtz-type decomposition at the discrete level and plays a key role in deriving error decay rates for the kinematic pressure. We derive (in some cases optimal) error decay rates for both the velocity vector field and kinematic pressure, with the error for the kinematic pressure measured in an $\textit{ad hoc}$ norm informed by the projection framework. The $\textit{a priori}$ error analysis remains robust even under reduced regularity of the velocity vector field and the kinematic pressure, and illustrates how the interplay of boundary conditions and projection stability governs the accuracy of pressure approximations.

math.NA↗

arXiv2025

Numerical analysis of the stochastic Navier-Stokes equations

The developments over the last five decades concerning numerical discretisations of the incompressible Navier--Stokes equations have lead to reliable tools for their approximation: those include stable methods to properly address the incompressibility constraint, stable discretisations to account for convection dominated problems, efficient time (splitting) methods, and methods to tackle their nonlinear character. While these tools may successfully be applied to reliably simulate even more complex fluid flow PDE models, their understanding requires a fundamental revision in the case of stochastic fluid models, which are gaining increased importance nowadays. This work motivates and surveys optimally convergent numerical methods for the stochastic Stokes and Navier--Stokes equations that were obtained in the last decades. Furtheremore, we computationally illustrate the failure of some of those methods from the deterministic setting, if they are straight-forwardly applied to the stochastic case. In fact, we explain why some of these deterministic methods perform sub-optimally by highlighting crucial analytical differences between the deterministic and stochastic equations -- and how modifications of the deterministic methods restore their optimal performance if they properly address the probabilistic nature of the stochastic problem. Next to the numerical analysis of schemes, we propose a general benchmark of prototypic fluid flow problems driven by different types of noise to also compare new algorithms by simulations in terms of complexities, efficiencies, and possible limitations. The driving motivation is to reach a better comparison of simulations for new schemes in terms of accuracy and complexities, and to also complement theoretical performance studies for restricted settings of data by more realistic ones.

math.NA↗

arXiv2024

Reaching the equilibrium: Long-term stable approximations for stochastic non-Newtonian Stokes equations with transport noise

We propose and analyse a novel, fully discrete numerical algorithm for the approximation of the generalised Stokes system forced by transport noise -- a prototype model for non-Newtonian fluids including turbulence. Utilising the Gradient Discretisation Method, we show that the algorithm is long-term stable for a broad class of particular Gradient Discretisations. Building on the long-term stability and the derived continuity of the algorithm's solution operator, we construct two sequences of approximate invariant measures. At the moment, each sequence lacks one important feature: either the existence of a limit measure, or the invariance with respect to the discrete semigroup. We derive an abstract condition that merges both properties, recovering the existence of an invariant measure. We provide an example for which invariance and existence hold simultaneously, and characterise the invariant measure completely. We close the article by conducting two numerical experiments that show the influence of transport noise on the dynamics of power-law fluids; in particular, we find that transport noise enhances the dissipation of kinetic energy, the mixing of particles, as well as the size of vortices.

math.NA↗

arXiv2024

Mean Square Temporal error estimates for the 2D stochastic Navier-Stokes equations with transport noise

We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.

math.NA↗

arXiv2024

A class of space-time discretizations for the stochastic $p$-Stokes system

The main objective of the present paper is to construct a new class of space-time discretizations for the stochastic $p$-Stokes system and analyze its stability and convergence properties. We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows to relate lower moments of discrete maximal processes. We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance. Moreover, we present an example such that the resulting velocity approximation converges with rate $1/2$ in time and $1$ in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.

math.NA↗

arXiv2024

On Young regimes for locally monotone SPDEs

We consider the following SPDE on a Gelfand-triple $(V, H, V^*)$: $$ du(t)=A(t,u(t))dt+dI_t(u), \qquad u(0)=u_0\in H. $$ Given certain local monotonicity, continuity, coercivity and growth conditions of the operator $A:[0, T]\times V\to V^*$ and a sufficiently regular operator $I$ we establish global existence of weak solutions. In analogy to the Young regime for SDEs, no probabilistic structure is required in our analysis, which is based on a careful combination of monotone operator theory and the recently developed Besov rough analysis by Friz and Seeger. Due to the abstract nature of our approach, it applies to various examples of monotone and locally monotone operators $A$, such as the $p$-Laplace operator, the porous medium operator, and an operator that arises in the context of shear-thickening fluids; and operators $I$, including additive Young drivers $I_t(u) = Z_t-Z_0$, abstract Young integrals $I_t(u) = \int_0^t σ(u_s)d X_s$, and translated integrals $I_t(u) = \int_0^t b(u_s - w_s)ds$ that arise in the context of regularization by noise. In each of the latter cases, we identify corresponding noise regimes (i.e. Young regimes) that assure our abstract result to be applicable. In the case of additive drivers, we identify the Brownian setting as borderline, i.e. noises which enjoy slightly more temporal regularity are amenable to our completely pathwise analysis.

math.PR↗

arXiv2023

On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise

We study the surface quasi-geostrophic equation driven by a generic additive noise process $W$. By means of convex integration techniques, we establish existence of weak solutions whenever the stochastic convolution $z$ associated with $W$ is well defined and fulfills certain regularity constraints. Quintessentially, we show that the so constructed solutions to the non-linear equation are controlled by $z$ in a linear fashion. This allows us to deduce further properties of the so constructed solutions, without relying on structural probabilistic properties such as Gaussianity, Markovianity or a martingale property of the underlying noise $W$.

math.PR↗

arXiv2023

A pathwise regularization by noise phenomenon for the evolutionary $p$-Laplace equation

We study an evolutionary $p$-Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this non-linear problem.

math.AP↗

arXiv2023

Temporal regularity of symmetric stochastic $p$-Stokes systems

We study the symmetric stochastic $p$-Stokes system, $p \in (1,\infty)$, in a bounded domain. The results are two-folded. First, we show that in the context of analytically weak solutions the stochastic pressure -- related to non-divergence free stochastic forces -- enjoys almost $-1/2$ temporal derivatives on a Besov scale. Second, we verify that the velocity component~$u$ of strong solutions obeys $1/2$ temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient $V(\varepsilon u) = (κ+ |\varepsilon u|)^{(p-2)/2} \varepsilon u$, $κ\geq 0$, has $1/2$ temporal derivatives in a Nikolskii space.

math.AP↗

arXiv2022

An averaged space-time discretization of the stochastic $p$-Laplace system

We study the stochastic $p$-Laplace system in a bounded domain. We propose two new space-time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and $1/2$ convergence in time. Additionally, we provide a sampling algorithm to construct the necessary random input in an efficient way. The theoretical error analysis is complemented by numerical experiments.

math.NA↗

arXiv2022

On temporal regularity for strong solutions to stochastic $p$-Laplace systems

In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$ time differentiability in an expontential Besov-Orlicz space for the solution process $u$. Furthermore, we prove $1/2$ time differentiability of the nonlinear gradient $|\nabla u|^{\frac{p-2}{2}} \nabla u$ in a Nikolskii space.

math.AP↗

arXiv2020

The parabolic p-Laplacian with fractional differentiability

We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space-time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskii spaces and therefore cover situations when the (gradient of) the solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolution $h$ and $τ$. The theoretical error analysis is complemented by numerical experiments.

math.NA↗