Two-grid algorithm for two-dimensional linear Schrödinger equation by the mixed finite element method

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids

In this article, an <i>H</i>¹-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and <i>L</i>1 formula are applied to approximate integer derivative and Caputo fractional derivative with order 1/2, respectively, and <i>H</i>¹-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both <i>H</i>¹-norm and <i>L</i>²-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method.

Two-grid mixed finite element method for nonlinear hyperbolic equations

Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem

SUMMARYIn this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley &amp; Sons, Ltd.

In this article, we investigate a priori error estimates for the optimal control problems governed by elliptic equations using higher order variational discretization and mixed finite element methods. The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is not discreted. A priori error estimates for the higher order variational discretization and mixed finite element approximation of control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results.Mathematics Subject Classification 1991: 49J20; 65N30.

Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

Flow fields in a centrifugal fan for an indoor unit of an air-conditioner were calculated with finite element method-based large eddy simulation (LES) with the aim of predicting fan performance and aerodynamic noise in this study. The numerical simulation code employed throughout the LES was called FrontFlow/blue (FFB). We compared 10M grid [coarse grid] and 60M grid [fine grid] calculation results for investigation of influence of grid resolution. In the fine grid, the number of grid elements in blade-to-blade direction, and of region between the shroud and the bell mouth increased in particular. By calculating with the fine grid, calculated distributions of absolute velocities at blade exit reasonably agreed with experimental results. Because of this, maximum absolute velocity by fine grid near hub decreased as compared to those by coarse grid. Calculated sound pressure level by fine grid was therefore smaller than that by coarse grid, and the overestimation of sound pressure was suppressed by calculating with fine grid. This decrease of the absolute velocity was a first factor for the improvement of calculation accuracy. Moreover, number of captured streaks on the blade, hub, and shroud surfaces by fine grid increased as compared to those by coarse grid. As a result, size of streak by fine grid became smaller than that by coarse grid. Static pressure fluctuations by fine grid on the blade, hub, and shroud surfaces therefore reduced as compared to those by coarse grid. Aerodynamic noise was related to static pressure fluctuations according to Curle’s equation. This reduction of static pressure fluctuations was therefore a second factor for improvement of calculation accuracy.

We study the numerical behaviour and the relationships between some numerical methods derived from the Mixed Finite Element (MFE) formulation and from the Control Volume formulation using the Multi-Point Flux Approximation (MPFA) of Aavatsmark et al. (1996). All methods are locally mass conservative and are well suited to solve the flow problem for both steady state (elliptic case) an transient (parabolic case) on a general irregular grid with anisotropic and heterogeneous discontinuous coefficients. Hybridization of the standard MFE method allows to obtain a symmetric positive definite system which is not, in general, an M-matrix. Moreover, the mixed hybrid finite element method uses more unknowns (one per edge or face) than the finite volume formulation (one per element). On the other hand, for a general element grid, the MPFA method does not give a symmetric matrix of coefficients and monotonicity issues are known to arise for high aspect ratios combined with skewed of computational grids. In this work, we propose two variants of the MFE method: (i) the lumped-MFE method, based on the equivalence between MHFE method and the P1-nonconforming Galerkin method for the Laplace equation and (ii) the multipoint-MFE method based on Multipoint finite volume discretization using the framework of the modified mixed finite element space. We discuss connections between the different approaches and perform numerical experiments for both steady state and transient cases to compare (number of unknowns, matrix properties, condition number) and study the numerical behaviour of these methods (unphysical oscillations, CPU time).

Running a fine grid model with 107 - 109 of cells is possible using a supercomputer with 103 - 106 of CPUs but may not be always cost-effective. The most cost-effective way is to use a coarse grid model that is much smaller but with static/dynamic profiles very close to the fine grid model. This paper proposes a new layer optimization and upscaling method with the aim for creating a consistent coarse grid model. Unlike the industry's existing layer optimization and upscaling methods, the proposed method performs layer optimization and upscaling fully integrated with the Lorenz coefficient and curves (LCC). Coarse grid layers and their permeabilities are created by minimizing the difference between fine and coarse grid LCCs. The process consists of static and dynamic optimizations. The former is measured by LCC while the latter by pressure, GOR, and water-cut. A new LCC-based permeability upscaling method is developed to preserve the fine grid multiphase flow behaviors. A satisfactory coarse grid model is achieved when both static and dynamic criteria are met. The proposed method has been successfully applied to a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within and between the matrix and fracture/karst features. The reservoir drive mechanisms are fluid expansion, miscible gas injection and aquifer drive. The reservoir is undersaturated and has an abnormally high initial reservoir pressure. The fine-grid static model contains 104 million cells (370×225×625×2) and the optimized upscaled coarse-grid dynamic model has 8.3 million cells (370×225×50×2). The upscaled model can be run efficiently on the company's existing HPC infrastructure with a maximum of 64 CPUs. Excellent matches of the Lorenz coefficient maps for reservoir total/zones and Lorenz curves at all wells between the fine and coarse grid models have been achieved. Matches on the dynamic variables, e.g., pressure, gas breakthrough time, and GOR growth, in all producers are within the defined acceptable tolerances. The high quality of the static and dynamic matches between the coarse- and fine-grid models confirms that the reservoir properties of the coarse-grid model is very close to the fine-grid model and can be used a base model for history matching and uncertainty analysis.

A Nitsche mixed extended finite element method for the biharmonic interface problem

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.