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Register a new userStability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media
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The purposes of this work are to study the $L^{2}$-stability of a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng in 1989 and to develop a Lagrange-Galerkin scheme with the Adams-Bashforth method to solve that model numerically.The stability estimate is obtained thanks to the presence of a nonlinear drag force term in the model which corresponds to the Forchheimer term. We derive the Lagrange-Galerkin scheme by extending the idea of the method of characteristics to overcome the difficulty which comes from the non-homogeneous porosity. Numerical experiments are conducted to investigate the experimental order of convergence of the scheme. For both simple and complex designs of porosities, our numerical simulations exhibit natural flow profiles which well describe the flow in non-homogeneous porous media.
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We consider an efficient preconditioner for boundary integral equation (BIE) formulations of the two-dimensional Stokes equations in porous media. While BIEs are well-suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as GMRES. In this paper, we apply a fast inexact direct solver, the inverse fast multipole method (IFMM), as an efficient preconditioner for GMRES. This solver is based on the framework of $mathcal{H}^{2}$-matrices and uses low-rank compressions to approximate certain matrix blocks. It has a tunable accuracy $varepsilon$ and a computational cost that scales as $mathcal{O} (N log^2 1/varepsilon)$. We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block-diagonal preconditioner, especially for pipe flow problems involving many pores.
Simulation of contact mechanics in fractured media is of paramount important in the scope of computational mechanics. In this work, a preconditioned mixed-finite element scheme with Lagrange multipliers is proposed in the framework of constrained variational principle, which has the capability to handle frictional contact mechanics of the multi-crossing fractures. The slippage, opening and contact traction on fractures are calculated by the resulted saddle-point algebraic system. A novel treatment is devised to guarantee physical solutions at the intersected position of crossing fractures. A preconditioning technique is introduced to re-scale the resulting saddle-point algebraic system, to preserve the robustness of the system. An iteration strategy, namely monolithic-updated contact algorithm, is then designed to update the two primary unknowns (displacement and Lagrange multiplier) in one algebraic block. A series of numerical tests is conducted to study the contact mechanics of single- and multi-crossing fractures. Benchmark study is presented to verify the presented numerical method. Two tests with crossing fractures are studied, in which the slippage and opening can be calculated. The effects of crossing fractures on the deformation field can be observed in the calculated results, in which the variation of slippage/opening is analyzed by different loading conditions.
In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media. The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver. To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations. Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains. The iterative and sequential IMplicit Pressure and Explicit Saturation (IMPES) algorithms are treated in time. Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG.
A mass-conservative Lagrange--Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange--Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-conservation property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-conservative scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $ell^infty(L^2)$- and $ell^2(H^1_0)$-norms. For the time increment $Delta t$, the mesh size $h$ and a conforming finite element space of polynomial degree $k$, the convergence order is of $O(Delta t^2 + h^k)$ in the $ell^infty(L^2)cap ell^2(H^1_0)$-norm and of $O(Delta t^2 + h^{k+1})$ in the $ell^infty(L^2)$-norm if the duality argument can be employed. Error estimates of $O(Delta t^{3/2}+h^k)$ in discre
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.
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