arXiv subjects

Aleksandra Zimmermann

Publications and source records attributed to Aleksandra Zimmermann.

16 recordsLinked to original sources

arXiv2025

Study of a TPFA scheme for the stochastic Allen-Cahn problem with constraint through numerical experiments

This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared $L^2$-error w.r.t. time is of order $1$ as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse.

math.NA↗

arXiv2025

Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraint

The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of Itô type. The problem is set in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and homogeneous Neumann boundary conditions are considered. The employed strategy consists in building a numerical scheme on a regularized version à la Moreau-Yosida of the constrained problem, and passing to the limit simultaneously with respect to the regularization parameter and the time and space steps, denoted respectively by $ε$, $Δt$ and $h$. Combining a semi-implicit Euler-Maruyama time discretization with a Two-Point Flux Approximation (TPFA) scheme for the spatial variable, one is able to prove, under the assumption $Δt=\mathcal{O}(ε^{2+θ})$ for a positive $θ$, the convergence of such a $(ε, Δt, h)$ scheme towards the unique weak solution of the initial problem, \textit{ a priori} strongly in $L^2(Ω;L^2(0,T;L^2(Λ)))$ and \textit{a posteriori} also strongly in $L^{p}(0,T; L^2(Ω\times Λ))$ for any finite $p\geq 1$.

math.NA↗

arXiv2025

Stochastic pseudomonotone parabolic obstacle problem: well-posedness $\&$ Lewy-Stampacchia's inequalities

We consider obstacle problems for nonlinear stochastic evolution equations. More precisely, the leading operator in our equation is a nonlinear, second order pseudomonotone operator of Leray-Lions type. The multiplicative noise term is given by a stochastic integral with respect to a Q-Wiener process. We show well-posedness of the associated initial value problem for random initial data on a bounded domain with a homogeneous Dirichlet boundary condition. First, we consider a singular perturbation of our problem by a higher order operator. Through the a priori estimates for the approximate solutions of the singular perturbation, only weak convergence is obtained. This convergence is not compatible with the nonlinearities in the equation. Therefore we use the theorems of Prokhorov and Skorokhod to establish existence of martingale solutions. Then, path-wise uniqueness follows from a L1-contraction principle and we may apply the method of Gyöngy-Krylov to obtain stochastically strong solutions. These well-posedness results serve as a basis for the study of variational inequalities and Lewy-Stampacchia's inequalities for our problem.

math.PR↗

arXiv2025

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function

We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the Itô's formula in Orlicz spaces. This general result applies to the ones studied in the literature, such as p(t, x)-Laplacian and double phase problems.

math.AP↗

arXiv2025

Convergence rates for a finite volume scheme of the stochastic heat equation

In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.

math.NA↗

arXiv2025

Well-posedness of a time discretization scheme for a stochastic p-Laplace equation with Neumann boundary conditions

In this contribution, we are interested in the analysis of a semi-implicit time discretization scheme for the approximation of a parabolic equation driven by multiplicative colored noise involving a $p$-Laplace operator (with $p\geq 2$), nonlinear source terms and subject to Neumann boundary conditions. Using the Minty-Browder theorem, we are able to prove the well-posedness of such a scheme.

math.AP↗

arXiv2024

A nonlinear stochastic diffusion-convection equation with reflection

We study a nonlinear, pseudomonotone, stochastic diffusion-convection evolution problem on a bounded spatial domain, in any space dimension, with homogeneous boundary conditions and reflection. The additive noise term is given by a stochastic Itô integral with respect to a Hilbert space valued $Q$-Wiener process. We show existence of a solution to the pseudomonotone stochastic diffusion-convection equation with non-negative initial value as well as the existence of a reflection measure which prevents the solution from taking negative values. In order to show a minimality condition of the measure, we study the properties of quasi everywhere defined representatives of the solution with respect to parabolic capacity.

math.AP↗

arXiv2024

Well-posedness of stochastic evolution equations with Hölder continuous noise

We show existence and pathwise uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution problem on a bounded domain $D\subseteq\mathbb{R}^d$, $d\in\mathbb{N}$, with homogeneous Dirichlet boundary conditions and random initial data $u_0\in L^2(Ω;L^2(D))$. The main novelty is the presence of a merely Hölder continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the Hölder noise term by inf-convolution and add a perturbation by a higher order operator to the equation. Using a stochastic compactness argument we may pass to the limit and we obtain first a martingale solution. Then by a pathwise uniqueness argument we get existence of a probabilistically strong solution.

math.PR↗

arXiv2024

On a finite-volume approximation of a diffusion-convection equation with a multiplicative stochastic force

We address an original approach for the convergence analysis of a finite-volume scheme for the approximation of a stochastic diffusion-convection equation with multiplicative noise in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and with homogeneous Neumann boundary conditions. The idea behind our approach is to avoid using the stochastic compactness method. We study a numerical scheme that is semi-implicit in time and in which the convection and the diffusion terms are respectively approximated by means of an upwind scheme and the so called two-point flux approximation scheme (TPFA). By adapting well-known methods for the time discretization of stochastic PDEs and combining them with deterministic techniques applied to spatial discretization, we show strong convergence of our scheme towards the unique variational solution of the continuous problem in $L^p(0,T;L^2(Ω;L^2(Λ)))$, for any finite $p\geq 1$.

math.NA↗

arXiv2023

Finite Volume Approximations for Non-Linear Parabolic Problems with Stochastic Forcing

We propose a two-point flux approximation finite-volume scheme for a stochastic non-linear parabolic equation with a multiplicative noise. The time discretization is implicit except for the stochastic noise term in order to be compatible with stochastic integration in the sense of Itô. We show existence and uniqueness of solutions to the scheme and the appropriate measurability for stochastic integration follows from the uniqueness of approximate solutions.

math.NA↗

arXiv2022

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem.

math.AP↗

arXiv2022

The stochastic $p$-Laplace equation on $\mathbb{R}^d$

We show well-posedness of the $p$-Laplace evolution equation on $\mathbb{R}^d$ with square integrable random initial data for arbitrary $1<p<\infty$ and arbitrary space dimension $d\in\mathbb{N}$. The noise term on the right-hand side of the equation may be additive or multiplicative. Due to a lack of coercivity of the $p$-Laplace operator in the whole space, the possibility to apply well-known existence and uniqueness theorems in the classical functional setting is limited to certain values of $1<p<\infty$ and also depends on the space dimension $d$. We propose a framework of functional spaces which is independent of Sobolev space embeddings and space dimension. For additive noise, we show existence using a time discretization. Then, a fixed-point argument yields the result for multiplicative noise.

math.PR↗

arXiv2021

Renormalized solutions for stochastic $p$-Laplace equations with $L^1$-initial data: The multiplicative case

We consider a $p$-Laplace evolution problem with multiplicative noise on a bounded domain $D \subset \mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p< \infty$. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic $p$-Laplace equations with $L^1$-initial data and study existence and uniqueness of solutions in this framework.

math.AP↗

arXiv2019

Well-posedness for the coupling of a random heat equation with a multiplicative stochastic Barenblatt equation

In this contribution, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a random heat equation coupled with a Barenblatt's type equation with a multiplicative stochastic force in the sense of Itô. In a first step we establish well-posedness in the case of an additive noise through a semi-implicit time discretization of the system. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case.

math.AP↗

arXiv2019

Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data

We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

math.AP↗

arXiv2014

Multi-dimensional scalar balance laws with discontinuous flux

We consider the problem of existence of entropy weak solutions to scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect both to the space variable x and the unknown quantity u. The problem is formulated in the framework of multi-valued mappings. We use the notion of entropy-measure valued solutions to prove the so-called contraction principle and comparison principle.

math.AP↗