Darcy Problem Research Articles

A penalty method for approximation of the stationary Stokes–Darcy problem

In this work, the penalty method is studied for the mixed Stokes–Darcy problem, motivated by the penalty method applied to Stokes equation. This work first proposes the penalty Stokes–Darcy model at the continuous level. Then we prove that the solution of the penalty model converges strongly to the original solution as Oϵ in which the penalty parameter is ϵ→0. What is more, the finite element method is used to solve the penalty model and the optimal error estimates are presented. Finally, several numerical tests are carried out to verify our theoretical results.

Optimal long time error estimates of a second-order decoupled finite element method for the Stokes–Darcy problem

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δt≤C(physical parameters), the optimal long time error estimates are obtained both in the L2 norm and in the H1 norm. Numerical results are provided to illustrate the convergence rate O(Δt2) of the temporal approximation.

Soret effect on stability of Darcy problem in a bidispersive porous medium with an exothermic chemical surface reaction

AbstractWe examine a saturated bidispersive porous medium undergoing an exothermic chemical reaction at its lower boundary. The fluid flow within this medium is modelled using the Darcy approach. While the Dufour effect is disregarded, the influence of the Soret effect is thoroughly explored. This problem represents a complex multiphysical phenomenon that integrates fluid dynamics, heat and mass transfer, chemical kinetics and thermoelectric effects. This problem finds relevance in various scientific and engineering applications, including enhanced oil recovery, chemical engineering, environmental engineering and energy systems. We introduce a complementary energy theory alongside a linear theoretical framework for the model. The Chebyshev collocation method is employed to derive both linear and nonlinear results. Our findings reveal that the effect of enhancing the Soret coefficient on the system's stability is contingent upon the boundary conditions and the Lewis number. Specifically, an increase in the Soret coefficient generally stabilises the system during stationary convection. Conversely, in scenarios where oscillatory convection prevails, a higher Soret coefficient tends to destabilise the system.

A hybrid Decoder-DeepONet operator regression framework for unaligned observation data

Deep neural operators (DNOs) have been utilized to approximate nonlinear mappings between function spaces. However, DNOs are confronted with challenges stemming from expanded dimensionality and computational costs tied to unaligned observation data, which ultimately compromise the accuracy of predictions. In this study, we present a hybrid Decoder-DeepONet framework to effectively handle unaligned data. This framework is advanced through its extension to the Multi-Decoder-DeepONet, which leverages an average field to enhance input augmentation. Furthermore, on the basis of the universal approximation theorem, we demonstrate that these frameworks preserve consistencies with operator approximation theory despite the substitution of the product with a decoder net. Two numerical experiments, Darcy problem and flow-field around an airfoil, are conducted to demonstrate the advantages of the proposed methods over conventional DeepONet approaches. The results reveal that both Decoder-DeepONet and Multi-Decoder-DeepONet utilize more compact training data dimensions and occupy less space, markedly enhancing prediction accuracy in the context of unaligned data.

Computation of the effective permeability of 2D doubly porous materials with elliptical shaped pores by using boundary element method

In recent years, the prediction of the effective transport properties have received a great number of investigations. The present work is dedicated to determining the effective permeability of two-dimensional (2D) doubly porous materials made of an isotropic permeable solid matrix in which elliptical shaped pores of any size are embedded. At the interface between the fluid and the solid, the Beaver–Joseph–Saffman conditions are applied. To achieve this objective, the Boundary Element Method (BEM) is first elaborated in the simulation of velocity and pressure solution fields of two coupled Stokes and Darcy problems. Afterwards, with the help of this solution results, the effective permeablity of the doubly porous material under investigation can be determined. For the purpose of assessing the accuracy and convergence of the BEM solution, the results obtained for the velocity and pressure fields are compared with the ones provided by the finite element method (FEM). Finally, several numerical examples are carried out to analyze the fluid/solid interface influence, the effect of area fraction and geometrical properties of pores, such as the size and distribution of the pores within the matrix phase.

A strongly conservative hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem

We present a strongly conservative and pressure-robust hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem. We show existence and uniqueness of a solution and present an optimal a priori error analysis for the fully discrete problem when using Backward Euler time stepping. The theoretical results are verified by numerical examples.

A DIMENSION-ADAPTIVE COMBINATION TECHNIQUE FOR UNCERTAINTY QUANTIFICATION

We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic partial differential equation, where the diffusion coefficient is parametrized by means of a Karhunen-Loève expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization, and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Loève coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm. For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.

The analogue of grad-div stabilization in DG method for a coupled Stokes and Darcy problem

The analogue of grad-div stabilization in DG method for a coupled Stokes and Darcy problem

Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling

Abstract This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

Variational inference in neural functional prior using normalizing flows: application to differential equation and operator learning problems

Physics-informed deep learning has recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the over-parameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In paper “MENG, X., YANG, L., MAO, Z., FERRANDIS, J. D., and KARNIADAKIS, G. E. Learning functional priors and posteriors from data and physics. Journal of Computational Physics, 457, 111073 (2022)”, a Bayesian framework based on the generative adversarial networks (GANs) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in “MENG, X., YANG, L., MAO, Z., FERRANDIS, J. D., and KARNIADAKIS, G. E. Learning functional priors and posteriors from data and physics. Journal of Computational Physics, 457, 111073 (2022)” has two stages: (i) prior learning, and (ii) posterior estimation. At the first stage, the GANs are utilized to learn a functional prior either from a prescribed function distribution, e.g., the Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference (VI), which naturally enables the mini-batch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional (100D) Darcy problem, are conducted to demonstrate that the NFs with full-/mini-batch training are able to achieve similar accuracy as the “gold rule” HMC. Moreover, the mini-batch training of NF makes it a promising tool for quantifying uncertainty in solving the high-dimensional partial differential equation (PDE) problems with big data.

Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L∞(Ω), where Ω⊂Rd is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in Lq(Ω), where 1≤q≤∞, using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space.

On the Conditions at the Interface Between Homogeneous and Porous Environments for the Stokes–Darcy Problem

On the Conditions at the Interface Between Homogeneous and Porous Environments for the Stokes–Darcy Problem

A hybridizable discontinuous Galerkin method for the coupled Navier–Stokes and Darcy problem

We present and analyze a strongly conservative hybridizable discontinuous Galerkin finite element method for the coupled incompressible Navier–Stokes and Darcy problem with Beavers–Joseph–Saffman interface condition. An a priori error analysis shows that the velocity error does not depend on the pressure, and that velocity and pressure converge with optimal rates. These results are confirmed by numerical examples.

Decoupling time filter method for the non‐stationary Navier‐Stokes/Darcy model

In this paper, we propose time filter algorithm based on the Backward Euler method for the non‐stationary Navier‐Stokes/Darcy problems. The decoupling method is composed of Backward Euler method and normal time filter. Stability and the second‐order convergence are also proved in details. Finally, numerical examples are listed to validate the theoretical analysis.

Robust Monolithic Solvers for the Stokes--Darcy Problem with the Darcy Equation in Primal Form

We construct mesh-independent and parameter-robust monolithic solvers for the coupled primal Stokes-Darcy problem. Three different formulations and their discretizations in terms of conforming and non-conforming finite element methods and finite volume methods are considered. In each case, robust preconditioners are derived using a unified theoretical framework. In particular, the suggested preconditioners utilize operators in fractional Sobolev spaces. Numerical experiments demonstrate the parameter-robustness of the proposed solvers.

On the convergence of the primal hybrid finite element method on quadrilateral meshes

We consider the approximation of a second order elliptic equation (Darcy problem) by using the Primal Hybrid Finite Element Method on quadrilateral meshes. We present new results in terms of sufficient, and in some cases also necessary, conditions to obtain the optimal convergence rates on convex quadrilaterals obtained from bilinear mappings. Numerical experiments are performed to illustrate the theoretical results.

Bend 3d mixed virtual element method for Darcy problems

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings.

Local and parallel efficient BDF2 and BDF3 rotational pressure-correction schemes for a coupled Stokes/Darcy system

In this paper, the local and parallel two- and three-step backward differentiation formula (BDF2/BDF3) rotational pressure-correction schemes are developed for a coupled Stokes/Darcy system. The central advantage of these schemes is a time-dependent version of domain decomposition by solving the Stokes problem and Darcy problems in their respective domain. By following a similar idea in Guermond et al. (2005), the Stokes problem is solved by a vector-valued elliptic equation and a scalar Poisson equation per time step. The whole system can be composed of three simple linear equations that consume almost the same computational time. Thus, the presented methods can be efficiently applied with less communication requirements and has good parallelism. In theorey, we prove the unconditional stability and long-time stability of the BDF2/BDF3 rotational pressure-correction schemes for the coupled Stokes/Darcy system. Furthermore, some numerical experiments are presented to show the accuracy and efficiency of these schemes in terms of numerical convergence rates and reservoir engineering.