Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAdaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization
119
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to save computational cost for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents any spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport.
rate research
Read More
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.
In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.
Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear--Scheidegger diffusion-dispersion tensor: $$ D({bf u}) = gamma d_m I + |{bf u}|bigg( alpha_T I + (alpha_L - alpha_T) frac{{bf u} otimes {bf u}}{|{bf u}|^2}bigg) , . $$ Previous works on optimal-order $L^infty(0,T;L^2)$-norm error estimate required the regularity assumption $ abla_xpartial_tD({bf u}(x,t)) in L^infty(0,T;L^infty(Omega))$, while the Bear--Scheidegger diffusion-dispersion tensor is only Lipschitz continuous even for a smooth velocity field ${bf u}$. In terms of the maximal $L^p$-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in $L^p(0,T;L^q)$-norm and almost optimal error estimate in $L^infty(0,T;L^q)$-norm are established under the assumption of $D({bf u})$ being Lipschitz continuous with respect to ${bf u}$.
In this paper, we study the stability and convergence of a decoupled and linearized mixed finite element method (FEM) for incompressible miscible displacement in a porous media whose permeability and porosity are discontinuous across some interfaces. We show that the proposed scheme has optimal-order convergence rate unconditionally, without restriction on the grid ratio (between the time-step size and spatial mesh size). Previous works all required certain restrictions on the grid ratio except for the problem with globally smooth permeability and porosity. Our idea is to introduce an intermediate system of elliptic interface problems, whose solution is uniformly regular in each subdomain separated by the interfaces and its finite element solution coincides with the fully discrete solution of the original problem. In order to prove the boundedness of the fully discrete solution, we study the finite element discretization of the intermediate system of elliptic interface problems.
We propose a Discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element $K$, a residual term involving the fluxes, measured in the norm of the dual of $H^1(K)$. The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the broken $H^1$ norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
