Postprocessing mixed finite element methods for the Cahn–Hilliard equation: the fully discrete case

Accurate simulation of Darcy flux is essential to simulate contaminant transport in groundwater accurately. The mixed finite element method has been used to obtain highly accurate flux distribution in groundwater flow applications. However, the method has not been widely adopted because of the lack of understanding of its merits, lack of comparison of its solution to those obtained from conventional schemes, and the usually mathematically rigorous presentation of the theory behind the method, which is not easily comprehensible. Hence, the objective of this paper is to present a simplified conceptual description of the mixed finite element method, to compare the solutions obtained from the method to those of the conventional finite element methods and to analyze any special properties of the solutions obtained from the mixed finite element methods. It has been shown in this paper that the solutions obtained from the mixed finite element method are highly accurate in rapidly changing flux distributions and heterogeneous flow distributions even when coarse grids are used to obtain the solution.

Mixed finite element methods (MFEM) form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order. They become increasingly important for the solution of nonlinear problems. In contrast to standard finite element schemes the mixed finite element discretisation of problems in divergence form, i.e. f +{\rm div} \, \sigma = 0 where \sigma = A (\nabla \, u) , \sigma \in L and u \in H , allows more flexibility in the design of the discrete approximation spaces contained in L and H , i.e. in the spaces for the direct variables and the Lagrange multipliers. The workshop focuses on new developments in the field of mixed and non-standard finite element methods. The main points are The workshop aimed at bridging the gap between the computational engineering community and applied mathematicians and in consequence to unify the scientific language and foster later collaboration. Nonlinear mixed schemes were of particular concern for problems in elasticity and plasticity, but electromagnetics and mathematically related topics were also included. Mixed finite element methods for elliptic problems are based on a variational description in saddle-point form. Side conditions such as divergence free velocity fields in incompressible fluid dynamics are usually treated in this framework. The appearance of ‘soft’ side conditions is typical for structural mechanics as is the case with nearly incompressible materials or plates and shells with small thickness parameters. We also mention materials which almost satisfy the Kirchhoff condition, i.e. problems with a high but finite shear stiffness. In such cases, which are by no means ‘soft’ from the mathematical point of view, mixed methods lead to a more robust discretisation. The arising stability conditions and computational techniques cannot be understood fully by intuitive mechanical principles; however, from the mathematician's point of view their reasoning is natural, clear and insightful. Mixed and non-standard finite element methods gain increasing prominence in the prevention of locking phenomena. We highlight a topic which is currently actively investigated: the development of stable and efficient plate and shell elements with regard to shear locking, which is more intricate than volume locking. Here it is important to understand how techniques based on heuristic ideas are consistent with more modern mathematical methods. Availability of fast solvers is decisive for the competitiveness of numerical techniques. For a variety of applications, multigrid methods are crucial for the efficiency of the implementation. Methods have been proposed which do not appear plausible if one wants to deduce the algorithms directly from the physical model. The advanced methods depend on rigorous error estimators in order to guarantee that the numerical solutions represent the exact solutions of the physical model.

Abstract: Groundwater flow modelling is of interest in many sciences and engineering applications for scientific understanding and/or technological management. Accurate numerical simulation of infiltration in the vadose zone remains a challenge, especially when very sharp fronts are present. This study is focused principally on an alternatively numerical approaches referred to in the literature as the mixed hybrid finite element (MHFE) method. MHFE schemes simultaneously approximate both the pressure head and its gradient. For some problems of unsaturated water flow, the MHFE solutions contain oscillations. Various authors ( see [1]) suggest the use of a mass lumping procedure to avoid this unphysical phenomenon. An analyse of the resulting matrix system shows that the recommended technique differs from the standard mass-lumping well-established for Galerkin finite element methods. A “new” effective mass-lumping scheme adapted from [2] has been specially developed for the MHFE method. Its ability for eliminating oscillations have been tested in unsaturated conditions. Various test cases in a 2D domain, for homogeneous and heterogeneous dry porous media and subject to different boundary conditions are presented.

A comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics (MMOC) combined with the Galerkin finite element method for the transport equation and the mixed finite element (MFE) method for the groundwater flow equation. The preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the MMOC and MFE discretizations. The MMOC takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. The physical diffusion and dispersion terms are treated by a standard finite element scheme. The crucial aspect of the MMOC technique is that it looks backward in time, along an approximate flow path, instead of forward in time as in many method of characteristics or moving mesh techniques. The MFE procedure involves solving for both the hydraulic head and the specific discharge simultaneously. One order of convergence is gained by the MFE method, as compared with other standard finite element methods, and therefore more accurate velocity fields are simulated. The overall advantages of the MMOC‐MFE method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations, and material balance errors are greatly reduced due to a very accurate velocity simulation by the MFE method. In addition, much larger time steps with Courant number well in excess of 1, as compared with the standard Galerkin finite element method, can be taken on a fixed spatial grid system without significant loss of accuracy.

Higher and lowest order mixed finite element approximation of subsurface flow problems with solutions of low regularity

Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients

We consider the lowest-order Raviart-Thomas mixed finite element method for second- order elliptic problems on simplicial meshes in two and three space dimensions. This method produces saddle-point problems for scalar and flux unknowns. We show how to easily and locally eliminate the flux unknowns, which implies the equivalence between this method and a particular multi-point finite volume scheme, without any approximate numerical integration. The matrix of the final linear system is sparse, positive definite for a large class of problems, but in general nonsymmetric. We next show that these ideas also apply to mixed and upwind-mixed finite element discretizations of nonlinear parabolic convection-diffusion-reaction problems. Besides the theoretical relationship between the two methods, the results allow for important computational savings in the mixed finite element method, which we finally illustrate on a set of numerical experiments.

The accuracy of the Darcy velocity, flux, and stream function computed from lowest‐order, triangle‐based, control volume and mixed finite element approximations to the two‐dimensional pressure equation is considered. The control volume finite element method, similar to integrated finite difference methods and analogous to the interpolation of Galerkin finite element results over “control volumes,” is shown to yield a conservative velocity field and smooth streamlines. The streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method, which approximates the pressure and velocity variables separately. It is shown that for systems with only moderate degrees of heterogeneity, the control volume finite element method is the more computationally efficient alternative; i.e., it provides more accurate flow results for a given number of unknowns. For more variable or discontinuous permeability fields, by contrast, such as sand/shale systems, the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns.

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

Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.

A nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium. A sequential implicit time‐stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a modified symmetric finite volume element method and the method of characteristics. Optimal order convergence in H1 and in L2 are proved for full discrete schemes. Finally, some numerical experiments are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Chapter 2 - Reduced-Order Extrapolation Finite Element Methods Based on Proper Orthogonal Decomposition