Efficient linear solvers for incompressible flow simulations using Scott‐Vogelius finite elements

An approximation of semiconductor device of heat conduction by mixed finite element method and characteristics-mixed finite element method

An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

Geometrically complex geological features, such as faults, cross stratified beds and large scale inclined strata, can have very significant effects on the flow of reservoir fluids. Standard finite difference methodology lacks the geometric flexibility required for modeling flow through or around such structures. Triangle based discretization methods, by contrast, offer an attractive means to discretize and simulate flow through geometrically complex features accurately. In this paper, a triangle based mixed finite element – finite volume technique for compositional reservoir simulation is developed. The formulation is of IMPES type and is applicable to three phase, multicomponent systems in two dimensions. Most previous mixed finite element formulations were restricted to two phase or fully miscible systems. The compositional approach presented here extends these earlier techniques though it retains the general structure of the more limited formulations. Specifically, the governing equations are cast in terms of a pressure equation and a series of component mass balance equations expressed in terms of the ‘total’ velocity. A triangle based mixed finite element method is applied to the solution of the pressure equation while a finite volume technique is employed for solution of the component mass balance equations. The overall methodology is particularly well suited for the modeling of flow through complex geological strata characterized by heterogeneous, generally anisotropic permeability fields. This is demonstrated explicitly through application of a preliminary implementation of the method to the simulation of flow through a faulted cross section with a complex permeability description. Extension of the formulation to the three dimensional case is also discussed.

A positive semi-definite problem of three-dimensional incompressible contamination treatment from nuclear waste in porous media is discussed in this paper. The mathematical model is defined by a nonlinear initial-boundary system consisting of partial differential equations. Four important equations (an elliptic equation, two convection-diffusion equations and a heat conductor equation) determine the physical features. Considering the physical natures and computational efficiency, the authors introduce the conservative mixed finite volume element, upwind approximation and multistep difference to solve this system. The pressure and Darcy velocity are computed by a mixed finite volume element. The concentrations and temperature are solved by a combination of upwind approximation, multistep difference and mixed finite volume element. A multistep difference is used for approximating the partial derivative with respect to time. Mixed finite volume element and upwind differences are given for solving the convection-diffusions equations. Numerical dispersion and nonphysical oscillations could be eliminated, and the computational efficiency is improved by using a large time step. Furthermore, a conservative law is preserved and error estimates in $L^2$-norm is obtained. Finally, two numerical experiments are given to show the efficiency and possible applications.

The solution of large sparse unsymmetric linear systems is a critical and challenging component of semiconductor device and circuit simulations. The time for a simulation is often dominated by this part. The sparse solver is expected to balance different, and often conflicting requirements. Reliability, a low memory-footprint, and a short solution time are a few of these demands. Currently, no black-box solver exists that can satisfy all criteria. The linear systems from both simulations can be highly ill-conditioned and are, therefore, quite challenging for direct and iterative methods. In this paper, it is shown that algorithms to place large entries on the diagonal using unsymmetric permutations and scalings greatly enhance the reliability of both direct and preconditioned iterative solvers for unsymmetric linear systems arising in semiconductor device and circuit simulations. The numerical experiments indicate that the overall solution strategy is both reliable and cost effective.

This article deals with the approximation of the bending of a clamped plate, modeled by Reissner‐Mindlin equations. It is known that standard finite element methods applied to this model lead to wrong results when the thickness t is small. Here, we propose a mixed formulation based on the Hellinger‐Reissner principle which is written in terms of the bending moments, the shear stress, the rotations and the transverse displacement. To prove that the resulting variational formulation is well posed, we use the Babuška‐Brezzi theory with appropriate t ‐dependent norms. The problem is discretized by standard mixed finite elements without the need of any reduction operator. Error estimates are proved. These estimates have an optimal dependence on the mesh size h and a mild dependence on the plate thickness t. This allows us to conclude that the method is locking‐free. The proposed method yields direct approximation of the bending moments and the shear stress. A local postprocessing leading to H1 ‐type approximations of transverse displacement and rotations is introduced. Moreover, we propose a hybridization procedure, which leads to solving a significantly smaller positive definite system. Finally, we report numerical experiments which allow us to assess the performance of the method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

ABSTRACTWe are interested in positivity preserving spatial discretizations applicable to arbitrary order mixed finite element spatial discretizations of the radiative diffusion equations. Though not guaranteed to yield strict positivity, matrix lumping is appealing as a method of increasing solution positivity because unlike ad hoc fixups or strictly non-negative, non-linear solution representations, matrix lumping keeps the mixed finite element diffusion equations as a linear system of equations. Self-lumping schemes have previously been shown to be a generalization of more traditional matrix lumping techniques. Self-lumping schemes restrict integration quadrature points to the interpolation points used to define interpolatory finite element basis functions. As shown previously, the positivity and order of accuracy of self-lumping schemes are directly related to the choice of finite element interpolation point. In this work, we develop self-lumping schemes for arbitrary order, mixed finite element spatial discretizations of the diffusion equation in Cartesian geometry. After introducing several possible self-lumping schemes appropriate for mixed finite elements, we test the ability of each scheme to increase solution positivity relative to an unlumped discretization using a simple test problem designed to induce negative solutions, evaluating efficacy as a function of finite element order and finite element interpolation point type. The method of manufactured solutions is then used to evaluate order of convergence for promising self-lumping schemes.

SummaryIn this exposition, we study both a priori and a posteriori error analysis for the H1‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and convergence properties for the state variables and the control variable are obtained. In addition, L2(L2)‐norm a posteriori error estimates for the state and control variables and ‐norm for the flux variable are also derived.

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).

The Scott–Vogelius element is a popular finite element for the discretization of the Stokes equations which enjoys inf-sup stability and gives divergence-free velocity approximations. However, it is well known that the convergence rates for the discrete pressure deteriorate in the presence of certain critical vertices in a triangulation of the domain. Modifications of the Scott–Vogelius element such as the recently introduced pressure-wired Stokes element also suffer from this effect. In this paper we introduce a simple modification strategy for these pressure spaces that preserves the inf-sup stability while the pressure converges at an optimal rate.

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures.

Hybrid finite element and modified mixed finite element solutions of axisymmetric circular plates

Faster energy convergence rate and more accurate calculation of the stress intensity factor KI have been reported, using the mixed finite element method, by Mirza and Olson (1978). However, the three node linear displacement and linear stress mixed element has five degrees of freedom (u, v, τxx τyy and τxy) per node and leads to a large half-bandwidth and consequently a very large computer storage requirement. Because of the superior performance of the mixed finite elements in linear elastic, stress singular problems it is desirable to use such elements near the crack tip. At the same time, away from the crack tip where stresses are distributed in a more regular manner, the conventional displacement finite elements may be employed. This leads to a transitional zone where the application of the Hellinger-Reissner mixed variational principle near the crack tip switches over to application of the potential energy theorem away from the crack tip. A transitional mixed finite element for connecting the linear displacement-linear stress (three node) mixed elements with the eight-node, displacement isoparametric elements have been developed by Leung (1982). This three element scheme leads to a much more efficient method computationally and yet maintains the energy convergence rate of 0(N−4) for both, the mixed and the eight node, isoparametric displacement elements. The energy release rate, using the equivalent of stiffness derivative technique by Park (1974), is used to calculate the stress intensity factor K•. Numerical calculations were found to be in very good agreement with the available results in the literature.

This study enhances the application of cross-sectional warping considered mixed finite element (WMFE) formulation to accurately determine natural vibration, static displacement response, and shear and normal stress evaluation with very close to the precision of solid finite elements (FEs) in two-phase/multi-phase functionally graded (FG) laminated composite beams strength using carbon nanotubes (CNTs). The principles of three dimensional (3D) elasticity theory are used to derive constitutive equations. The mixed finite element (MFE) method is improved by accounting for warping effects by displacement-based FEs within the cross-sectional domain. The MFE with two nodes has a total of 24 degrees of freedom. The two-phase material consists of a polymer matrix reinforced with aligned CNTs that are FG throughout the beam thickness. The multi-phase FG beam is modeled as a three-component composite material, consisting of CNTs, a polymer matrix, and fibers. The polymer matrix is reinforced by longitudinally aligned fibers and randomly dispersed CNT particles. The fiber volume fractions are considered to change gradually through the thickness of the beam following a power-law variation. The W-MFE achieves satisfactory results with fewer degrees of freedom than 3D solid FEs. Benchmark examples examine the effects of ply orientation, configuration, and fiber gradation on FG beam behavior.

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