Observability Inequalities of a Semi-discrete Schrödinger Integro-Differential Equation Derived from a Mixed Finite Element Method

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.

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.

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.

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

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.

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.

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

A combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is investigated. That is to say, the mixed finite element method with Raviart-Thomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin (SIPG) approximation. Based on projection interpolations and induction hypotheses, a superconvergence estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration.

Mixed and nonconforming finite element methods on a system of polygons

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.

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

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

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

Geological media exhibit permeability fields and porosities that differ by several orders of magnitude across highly varying length scales. Computational methods used to model flow through such media should be capable of treating rough coefficients and grids. Further, the adherence of these methods to basic physical properties such as local mass balance and continuity of fluxes is of great importance. Both discontinuous Galerkin (DG) and mixed finite element (MFE) methods satisfy local mass balance and can accurately treat rough coefficients and grids. The appropriate choice of physical models and numerical methods can substantially reduce computational cost with no loss of accuracy. MFE is popular due to its accurate approximation of both pressure and flux but is limited to relatively structured grids. On the other hand, DG supports higher order local approximations, is robust and handles unstructured grids, but is very expensive because of the number of unknowns. To this end, we present DG-DG and DG-MFE domain decomposition couplings for slightly compressible single phase flow in porous media. Mortar finite elements are used to impose weak continuity of fluxes and pressures on the interfaces. The sub-domain grids can be non-matching and the mortar grid can be much coarser making this a multiscale method. The resulting nonlinear algebraic system is solved via a non-overlapping domain decomposition algorithm, which reduces the global problem to an interface problem for the pressures. Solutions of numerical experiments performed on simple test cases are first presented to validate the method. Then, additional results of some challenging problems in reservoir simulation are shown to motivate the future application of the theory.