arXiv subjects

Piotr Bartman-Szwarc

Publications and source records attributed to Piotr Bartman-Szwarc.

4 recordsLinked to original sources

arXiv2026

Finite Element Approximation of a Hemivariational Inequality for Steady-State Heat Conduction: Double-Limit Convergence of Penalization and Discretization

In this paper we study the numerical approximation and convergence analysis of a steady-state heat conduction problem with mixed boundary conditions. The physical model is governed by a hemivariational inequality depending on a heat transfer parameter $α> 0$. We consider the finite element approximation of this penalized problem, as well as its corresponding limit problem with a prescribed constant temperature on a part of the boundary. The main theoretical result establishes the strong convergence of the discrete solutions. Specifically, we prove the double-limit convergence of the finite element approximations to the limit solution as the mesh size $h$ tends to zero and the penalization parameter $α$ tends to infinity, independently and simultaneously. We further complement the convergence analysis with an error estimate of optimal order for the discrete limit problem, and we explain why an estimate uniform in the penalization parameter cannot be expected. The theoretical results are illustrated by numerical simulations on four examples verified by the method of manufactured solutions.

math.NA↗

arXiv2025

Adaptive time-stepping for the Super-Droplet Method Monte Carlo collision-coalescence scheme

We present an analysis of an adaptive time-stepping scheme for the Super-Droplet Method (SDM), a Monte Carlo algorithm for simulating particle coagulation. SDM represents cloud droplets as weighted superdroplets, enabling high-fidelity representations of microphysical processes such as collision-coalescence. However, the algorithm can undercount collisions when the expected number of events is not realizable given the superdroplet configuration, introducing a biased error referred here as the collision deficit. While SDM exhibits statistical spread inherent to Monte Carlo schemes, the deficit is a systematic underestimation of collision events. This error can be addressed with adaptive time-stepping, which dynamically adjusts simulation time steps to eliminate this deficit. We analyze the behavior of the deficit across a wide range of timesteps, superdroplet counts, and initialization strategies, and explore trade-offs between accuracy and efficiency. Using the classical Safranov-Golovin test case, we show that the deficit increases with timestep and superdroplet count, and that adaptive time-stepping effectively removes the associated error without significant cost. We test a smooth continuum of initial distributions with extrema representing two different initialization methods, and find that while the deficit is sensitive to the choice of attribute-space sampling strategies, adaptive time-stepping substantially reduces the difference, allowing for users to choose initialization methods optimized for other processes. We also propose a method of visualization, capturing both the attribute sampling, droplet interactions over multiple timesteps, and the deficit using network connectivity graphs. In 2-D flow-coupled simulations, we find the deficit can have a stronger effect on convergence than previously shown, with uncorrected deficit delaying the onset of precipitation.

physics.ao-ph↗

arXiv2025

Hybrid subgradient and simulated annealing method for hemivariational inequalities

In this paper, we employ a global aggregate subgradient method for the numerical solution of hemivariational inequality problems arising in contact mechanics. The method integrates a global search procedure to identify starting points for a local minimization algorithm. The algorithm consists of two types of steps: null steps and serious steps. In each null step, only two subgradients are utilized: the aggregate subgradient and the subgradient computed at the current iteration point, which together determine the search direction. Furthermore, we compare the performance of the proposed method with selected solvers using a representative contact mechanics problem as a case study.

math.OC↗

arXiv2024

A new penalty method for elliptic quasivariational inequalities

We consider a class of elliptic quasivariational inequalities in a reflexive Banach space $X$ for which we recall a convergence criterion obtained in [10]. Each inequality $\cal P$ in the class is governed by a set of constraints $K$ and has a unique solution $u\in K$. The criterion provides necessary and sufficient conditions which guarantee that an arbitrary sequence $\{u_n\}\subset X$ converges to the solution $u$. Then, we consider a sequence $\{\cal P_n\}$ of unconstrained variational-hemivariational inequalities governed by a sequence of parameters $\{λ_n\}\subset\mathbb{R}_+$. We use our criterion to deduce that, if for each $n\in\mathbb{N}$ the term $u_n$ represents a solution of Problem $\cal P_n$, then the sequence $\{u_n\}$ converges to $u$ as $λ_n\to 0$. We apply our abstract results in the study of an elastic frictional contact problem with unilateral constraints and provide the corresponding mechanical interpretations. We also present numerical simulation in the study of a two-dimensional example which represents an evidence of our convergence results.

math.AP↗