arXiv subjects

Iuliu Sorin Pop

Publications and source records attributed to Iuliu Sorin Pop.

At least 19 recordsLinked to original sources

arXiv2026

An adaptive, space-time discretized linear iterative scheme for doubly-degenerate parabolic problems

Degenerate diffusion problems, where the governing parabolic equation can change type to either an ordinary differential equation or an elliptic equation, model many real life applications. Due to the presence of free-boundaries, accurate numerical simulation of such problems require extremely small mesh and time step sizes locally. To remediate this issue, in this work, we consider a space-time formulation of the problem based on an efficient splitting of the nonlinearities. First, an iterative linearization scheme is proposed to resolve the nonlinearities that effectively reduces to solving a sequence of heat equations. Unconditional convergence of the scheme is proven even for double degenerate cases with linear convergence achieved if the problem is non-degenerate. Next, the dual norm of the nonlinear residual is decomposed into a linearization error component and a discretization error component corresponding to the heat equation. This leads to reliable and fully computable a posteriori estimates for the problem that are robust with respect to the nonlinearities/degeneracies. These estimates are used then in a fully adaptive (discretization + linearization) space-time solver. Numerical experiments for multiple test cases (one and two dimensions in space) demonstrate that this solver efficiently allocates the computational resources in the space-time domain, resulting in a rapid decay of error in terms of total degrees of freedom spent.

math.NA↗

arXiv2025

Robust, fast, and adaptive splitting schemes for nonlinear doubly-degenerate diffusion equations

We consider linear iterative schemes for the time-discrete equations stemming from a class of nonlinear, doubly-degenerate parabolic equations. More precisely, the diffusion is nonlinear and may vanish or become multivalued for certain values of the unknown, so the parabolic equation becomes hyperbolic or elliptic, respectively. After performing an Euler implicit time-stepping, a splitting strategy is applied to the time-discrete equations. This leads to a formulation that is more suitable for dealing with the degeneracies. Based on this splitting, different iterative linearization strategies are considered, namely the Newton scheme, the L-scheme, and the modified L-scheme. We prove the convergence of the latter two schemes even for the double-degenerate case. In the non-degenerate case, we prove that the scheme is contractive, and the contraction rate is proportional to a non-negative exponent of the time-step size. Moreover, an a posteriori estimator-based adaptive algorithm is developed to select the optimal parameters for the M-scheme, which accelerates its convergence. Numerical results are presented, showing that the M- and the M-adaptive schemes are more stable than the Newton scheme, as they converge irrespective of the mesh. Moreover, the adaptive M-scheme consistently out-competes not only the M/L-schemes, but also the Newton scheme showing quadratic convergence behavior.

math.NA↗

arXiv2025

Efficient iterative linearised solvers for numerical approximations of stochastic Stefan problems

We present iterative solvers to approximate the solution of numerical schemes for stochastic Stefan problems. After briefly talking about the convergence results, we tackle the question of efficient strategies for solving the nonlinear equation associated with this scheme. We explore several approaches, from a standard Newton technique to linearised solvers. The latter offer the advantage of using the same coefficient matrix of the linearised system in each nonlinear iteration, for all time steps, and across all realisations of the Brownian motions. As a consequence, the system can be factorised once and for all. Although the linearised approach has a slower convergence rate, our sensitivity analysis and the use of adaptive tolerance in both deterministic and stochastic cases provide valuable insights for choosing the most effective solver across various scenarii.

math.NA↗

arXiv2024

Rigorous derivation of an effective model for coupled Stokes advection, reaction and diffusion with freely evolving microstructure

We consider the homogenisation of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size $\varepsilon$. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer having a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection-reaction-diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for $\varepsilon \to 0$, based on $\varepsilon$-uniform a priori estimates. As a result of the limit passage, the processes on the macroscale are described by an advection-reaction-diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells, which depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro-macro model. For pure reactive-diffusive transport coupled with microscopic domain evolution but without advective transport, homogenisation results have recently been presented. We extend these models by advective transport which is driven by the Stokes equation in the a priori unknown evolving pore domain.

math.AP↗

arXiv2022

Homogenization of a mineral dissolution and precipitation model involving free boundaries at the micro scale

In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. Hence, we are dealing with a multi-scale problem with free boundaries on the pore scale. In the initial state the microscopic geometry is given by a periodically perforated domain, including spherical solid grains. The radius of each grain is of order $ε$ and depends on the unknown (the solute concentration) at its surface. Therefore the radii of the grains change in time, leading to a nonlinear, free boundary problem. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to $ε$. Finally, letting $ε\to 0$, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.

math.AP↗

arXiv2021

Upscaling of a Cahn-Hilliard Navier-Stokes Model with Precipitation and Dissolution in a Thin Strip

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Peclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip is going to zero. In this way and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell-problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell-problems can be solved explicitly. Notably we find asymptotic consistency, that is the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.

math.AP↗

arXiv2020

Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media

We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.

math.NA↗

arXiv2020

An efficient numerical scheme for fully coupled flow and reactive transport in variably saturated porous media including dynamic capillary effects

In this paper, we study a model for the transport of an external component, e.g., a surfactant, in variably saturated porous media. We discretize the model in time and space by combining a backward Euler method with the linear Galerkin finite elements. The Newton method and the L-Scheme are employed for the linearization and the performance of these schemes is studied numerically. A special focus is set on the effects of dynamic capillarity on the transport equation.

math.NA↗

arXiv2020

Iterative schemes for surfactant transport in porous media

In this work we consider the transport of a surfactant in a variably saturated porous media. The water flow is modelled by the Richards equations and it is fully coupled with the transport equation for the surfactant. Three linearization techniques are discussed: the Newton method, the modified Picard and the L-scheme. Based on these, monolithic and splitting schemes are proposed and their convergence is analyzed. The performance of these schemes is illustrated on four numerical examples. For these examples, the number of iterations and the condition numbers of the linear systems emerging in each iteration are presented.

math.NA↗

arXiv2020

An adaptive multi-scale iterative scheme for a phase-field model for precipitation and dissolution in porous media

Mineral precipitation and dissolution processes in a porous medium can alter the structure of the medium at the scale of pores. Such changes make numerical simulations a challenging task as the geometry of the pores changes in time in an apriori unknown manner. To deal with such aspects, we here adopt a two-scale phase-field model, and propose a robust scheme for the numerical approximation of the solution. The scheme takes into account both the scale separation in the model, as well as the non-linear character of the model. After proving the convergence of the scheme, an adaptive two-scale strategy is incorporated, which improves the efficiency of the simulations. Numerical tests are presented, showing the efficiency and accuracy of the scheme in the presence of anisotropies and heterogeneities.

math.NA↗

arXiv2019

Mathematical Modeling, Laboratory Experiments, and Sensitivity Analysis of Bioplug Technology at Darcy Scale

In this paper we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by an extended version of Darcy's law and the substrate is transported by diffusion and convection in the water phase. Initially there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space which changes the rock permeability. The model includes detachment of biomass due to water flux and death of bacteria, and is implemented in MRST. We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multi-wavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity-permeability relations in a core saturated with water and oil.

physics.app-ph↗

arXiv2019

Dynamic and weighted stabilizations of the $L$-scheme applied to a phase-field model for fracture propagation

We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called $L$-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.

math.NA↗

arXiv2019

Global existence of a weak solution to unsaturated poroelasticity

In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biot's quasi-static consolidation model. The coupled, elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media obtained under similar assumptions as usually considered for Richards' equation. In this work, the existence of a weak solution is established using regularization techniques, the Galerkin method, and compactness arguments. The final result holds under non-degeneracy conditions and natural continuity properties for the non-linearities. The assumptions are demonstrated to be reasonable in view of geotechnical applications.

math.AP↗

arXiv2019

An upscaled model for permeable biofilm in a thin channel and tube

In this paper, we derive upscaled equations for modelling biofilm growth in porous media. The resulting macro-scale mathematical models consider permeable multi-species biofilm including water flow, transport, detachment and reactions. The biofilm is composed of extracellular polymeric substances (EPS), water, active bacteria and dead bacteria. The free flow is described by the Stokes and continuity equations and the water flux inside the biofilm by the Brinkman and continuity equations. The nutrients are transported in the water phase by convection and diffusion. This pore-scale model includes variations of the biofilm composition and size due to reproduction of bacteria, production of EPS, death of bacteria and shear forces. The model includes a water-biofilm interface between the free flow and the biofilm. Homogenization techniques are applied to obtain upscaled models in a thin channel and a tube, by investigating the limit as the ratio of the aperture to the length $\varepsilon$ of both geometries approaches to zero. As $\varepsilon$ gets smaller, we obtain that the percentage of biofilm coverage area over time predicted by the pore-scale model approaches the one obtained using the effective equations, which shows a correspondence between both models. The two derived porosity-permeability relations are compared to two empirical relations from the literature. The resulting numerical computations are presented to compare the outcome of the effective (upscaled) models for the two mentioned geometries.

physics.flu-dyn↗

arXiv2019

Fronts in two-phase porous flow problems: effects of hysteresis and dynamic capillarity

In this work, we study the behaviour of saturation fronts for two phase flow through a long homogeneous porous column. In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses travelling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include non-monotone profiles, multiple shocks and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we compare experimental results with our theoretical findings.

math.DS↗

arXiv2018

A pore-scale model for permeable biofilm: numerical simulations and laboratory experiments

In this paper we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided in two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substances (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation whereas a diffusion-convection equation is involved for the transport of nutrients. At the water/biofilm interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/dead of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by $\pm 10 \%$ of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.

physics.flu-dyn↗

arXiv2018

Upscaling of unsaturated flow in fractured porous media

In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called $\varepsilon > 0$. Then we take the limit $\varepsilon \to 0$ and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to $\varepsilon$, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

math.NA↗

arXiv2018

Linear iterative schemes for doubly degenerate parabolic equations

Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation. Here the backward Euler method is combined with a mixed finite element method scheme, which results in a stable and locally mass-conservative scheme. At the same time, at each time step one has to solve a non-linear algebraic system, for which linear iterations are needed. Finding robust and convergent ones is particularly challenging here, since both slow and fast diffusion cases are allowed. Commonly used schemes, like Newton and Picard iterations, are defined either for non-degenerate problems, or after regularising the problem in the case of degenerate ones. Convergence is guaranteed only if the initial guess is sufficiently close to the solution, which translates into severe restrictions on the time step. Here we discuss a linear iterative scheme which builds on the $L$-scheme, and does not employ any regularisation. We prove its rigourous convergence, which is obtained for mild restrictions on the time step. Finally, we give numerical results confirming the theoretical ones, and compare the behaviour of the scheme with other schemes.

math.NA↗