Multipoint flux mixed finite element methods for slightly compressible flow in porous media

In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity.     

Stochastic mixed multiscale finite element methods and their applications in random porous media

We present a framework for stochastic mixed multiscale finite element methods (mixed MsFEMs) for elliptic equations with heterogeneous random coefficients. The use of some global information is necessary in multiscale simulations when there is no scale separation for the heterogeneity. The methods in the proposed framework for the stochastic mixed MsFEMs use some global information. The media properties in a stochastic environment drastically vary among realizations and, thus, many global fields are needed for multiscale simulation. The computations of these global fields on a fine grid can be very expensive. One can utilize upscaling methods to compute the global information on an intermediate coarse grid that reduces the computational cost. We investigate two approaches of stochastic mixed MsFEMs in the framework. First approach entails no stochastic interpolation and the second approach uses stochastic interpolation. If the random media have deterministic features that play significant roles in the flow, we can use the deterministic features of the random media as the global information. This reduces the computational cost of the simulations. We make convergence analysis of the stochastic mixed MsFEMs and investigate their applications to incompressible two-phase flows in random porous media. The numerical results demonstrate the effectiveness of the proposed methods and confirm the convergence.     

A finite element approach for multiphase fluid flow in porous media

The main scope of this work is to carry out a mathematical framework and its corresponding finite element (FE) discretization for the partially saturated soil consolidation modelling in presence of an immiscible pollutant. A multiphase system with the interstitial voids in the grain matrix filled with water (liquid phase), water vapour and dry air (gas phase) and with pollutant substances, is assumed. The mathematical model addressed in this work was developed in the framework of mixture theory considering the pollutant saturation-suction coupling effects. The ensuing mathematical model involves equations of momentum balance, energy balance and mass balance of the whole multiphase system. Encouraging outcomes were achieved in several different examples.     

Fully discrete potential-based finite element methods for a transient eddy current problem

Fully discrete potential-based finite element methods called methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7. This work was supported by Postech BSRI Research Fund-2009, National Basic Research Program of China (2008CB425701), NSFC under the grant 10671025 and the Key Project of Chinese Ministry of Education (No. 107018).     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

A two-grid expanded mixed element method for nonlinear non-Fickian flow model in porous media

The full discrete scheme of expanded mixed finite element approximation is introduced for nonlinear parabolic integro-differential equations modelling non-Fickian flow in porous media. To solve the nonlinear problem efficiently, a two-grid algorithm is considered and analysed. This approach allows us to perform all of the nonlinear iterations on a coarse grid space and just execute a linear system on a fine grid space. Based on RT_k mixed element space, error estimates and convergence results are presented for solutions of the two-grid method. Some numerical examples are given to verify the theoretical predictions and show the efficiency of the two-grid method.     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

A parallel finite element scheme for thermo-hydro-mechanical (THM) coupled problems in porous media

Many applied problems in geoscience require knowledge about complex interactions between multiple physical and chemical processes in the sub-surface. As a direct experimental investigation is often not possible, numerical simulation is a common approach. The numerical analysis of coupled thermo-hydro-mechanical (THM) problems is computationally very expensive, and therefore the applicability of existing codes is still limited to simplified problems. In this paper we present a novel implementation of a parallel finite element method (FEM) for the numerical analysis of coupled THM problems in porous media. The computational task of the FEM is partitioned into sub-tasks by a priori domain decomposition. The sub-tasks are assigned to the CPU nodes concurrently. Parallelization is achieved by simultaneously establishing the sub-domain mesh topology, synchronously assembling linear equation systems in sub-domains and obtaining the overall solution with a sub-domain linear solver (parallel BiCGStab method with Jacobi pre-conditioner). The present parallelization method is implemented in an object-oriented way using MPI for inter-processor communication. The parallel code was successfully tested with a 2-D example from the international DECOVALEX benchmarking project. The achieved speed-up for a 3-D extension of the test example on different computers demonstrates the advantage of the present parallel scheme.     

Nodal finite element approximations for the neutron transport equation

Two classes of nodal methods, weakly and strongly discontinuous, are introduced and applied to the numerical solution of the neutron transport equation in two-dimensional Cartesian geometry and discrete ordinates. These methods are then applied for the approximation of the solution of a reference problem well known in the nuclear engineering literature.     

Reduced integration for improved accuracy of finite element approximations

For completeness the finite element bases which are used for approximate solutions of elliptic problems of order 2p by the Ritz method must include the functions corresponding to the constant value of the pth derivative. In actual usage, to ensure a positive definite system of algebraic equations, additional interpolating functions are introduced. This leads to “multiple covering” of some of the system modes and results in overestimation of stiffness. Reduced integration techniques eliminate some of this multiple covering and thereby give improved accuracy. Selective reduced integration has been found useful in the analysis of flexural problems. In this paper we suggest the use of only the minimal covering that is sufficient for convergence. A technique for solution of the discretized system is given. Numerical performance data show remarkable improvement over conventional procedures. The proposed scheme yields good approximation even for very coarse meshes. This indicates the possibility of considerable economy in the cost of obtaining finite element solutions to complex problems, e.g. coupled field problems, three-dimensional problems, stress concentration etc.     

A mixed virtual element method for a nonlinear Brinkman model of porous media flow

In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the \(\mathbb {L}^2\)-orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken \(\mathbb {H}(\mathbf {div})\)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.     

A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

Electric potential approximations for an eight node plate finite element

The aim of this work is to develop a computational tool for multilayered piezoelectric plates: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, two approaches were previously used for the mechanical part, taking into account the transverse shear stress effects and using only five unknown generalized displacements: C⁰ finite element approximation based on first-order shear deformation theories (FSDT) [Polit O, Touratier M, Lory P. A new eight-node quadrilateral shear-bending plate finite element. Int J Numer Meth Eng 1994;37:387–411] and C¹ finite element approximations using a high order shear deformation theory (HSDT) [Polit O, Touratier M. High order triangular sandwich plate finite element for linear and nonlinear analyses. Comput Meth Appl Mech Eng 2000;185:305–24]. In this article, we present the piezoelectric extension of the FSDT eight node plate finite element. The electric potential is approximated using the layerwise approach and an evaluation is proposed in order to assess the best compromise between minimum number of degrees of freedom and maximum efficiency. On one side, two kinds of finite element approximations for the electric potential with respect to the thickness coordinate are presented: a linear variation and a quadratic variation in each layer. On the other side, the in-plane variation can be quadratic or constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post-processing level, the transverse shear stresses are deduced using the equilibrium equations.Numerous tests are presented in order to evaluate the capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions. Finally, an adaptative composite plate is evaluated using the best compromise finite element.     

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

In this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. By applying some error estimates techniques of mixed finite element methods, we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.     

A 3D finite element model for free-surface flows

The present work deals with the validation of 3D finite element model for free-surface flows. The model uses the non-hydrostatic pressure and the eddy viscosities from the conventional linear turbulence model are modified to account for the secondary effects generated by strong channel curvature in the natural rivers with meandering open channels. The unsteady Reynolds-averaged Navier–Stokes equations are solved on the unstructured grid using the Raviart–Thomas finite element for the horizontal velocity components, and the common P1 linear finite element in the vertical direction. To provide the accurate resolution at the bed and the free-surface, the governing equations are solved in the multi-layers system (the vertical plane of the domain is subdivided into fixed thickness layers). The up-to-date k–ε turbulence solver is implemented for computing eddy coefficients, the Eulerian–Lagrangian–Galerkin (ELG) temporal scheme is performed for enhancing numerical time integration to guarantee high degree of mass conservation while the CFL restriction is eliminated. The present paper reports on successful validation of the numerical model through available benchmark tests with increasing complexity, using the high quality and high spatial resolution three-dimensional data set collected from experiments.     

A Newton-like finite element scheme for compressible gas flows

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense.     

Finite central difference/finite element approximations for parabolic integro-differential equations

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δt ² + h ^l ). A numerical example demonstrates the theoretical results.     

Adaptive finite volume methods for displacement problems in porous media

In this paper we consider adaptive numerical simulation of miscible and immiscible displacement problems in porous media, which are modeled by single and two phase flow equations. Using the IMPES formulation of the two phase flow equation both problems can be treated in the same numerical framework. We discretise the equations by an operator splitting technique where the flow equation is approximated by Raviart-Thomas mixed finite elements and the saturation or concentration equation by vertex centered finite volume methods. Using a posteriori error estimates for both approximation schemes we deduce an adaptive solution algorithm for the system of equations and show the applicability in several examples.