Mixed Finite Volume Methods Research Articles

Coupled mixed finite element and finite volume methods for a solid velocity-based model of multidimensional sedimentation

In this paper we introduce and analyze a model of sedimentation based on a solid velocity formulation. A particular feature of the governing equations is given by the fact that the velocity field is non-divergence free. We introduce extra variables such as the pseudostress tensor relating the velocity gradient with the pressure, thus leading to a mixed variational formulation consisting of two systems of equations coupled through their source terms. A result of existence and uniqueness of solutions is shown by means of a fixed-point strategy and the help of the Babuška–Brezzi theory and Banach theorem. Additionally, we employ suitable finite dimensional subspaces to approximate both systems of equations via associated mixed finite element methods. The well-posedness of the resulting coupled scheme is also treated via a fixed-point approach, and hence the discrete version of the existence and uniqueness result is derived analogously to the continuous case. The above is then combined with a finite volume method for the transport equation. Finally, several numerical results illustrating the performance of the proposed model and the full numerical scheme, and confirming the theoretical rates of convergence, are presented.

The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair

The Mixed Finite Volume Methods for Stokes Problem Based on MINI Element Pair

Compressible Navier–Stokes Equations with Potential Temperature Transport: Stability of the Strong Solution and Numerical Error Estimates

We present a dissipative measure-valued (DMV)-strong uniqueness result for the compressible Navier–Stokes system with potential temperature transport. We show that strong solutions are stable in the class of DMV solutions. More precisely, we prove that a DMV solution coincides with a strong solution emanating from the same initial data as long as the strong solution exists. As an application of the DMV-strong uniqueness principle we derive a priori error estimates for a mixed finite element-finite volume method. The numerical solutions are computed on polyhedral domains that approximate a sufficiently a smooth bounded domain, where the exact solution exists.

Research on the thermo-hydraulic response characteristics of pore-fissure media using mixed finite volume method

After lengthy diagenesis and tectonic movement, a rock mass inevitably develops many pores and micro-fissures. A numerical simulation method was employed to study the thermal response characteristics of the rock mass under temperature seepage coupling by treating it as a pore-fissure medium and considering its anisotropic properties. Based on the mixed finite volume method (FVM), a numerical scheme of the governing equation for the temperature seepage coupling of the pore-fissure medium is derived, with the program solution module independently written in C++. On this basis, a numerical test model of the fissured mudstone is established to analyze the distribution of the rock mass temperature field under various thermal conductivities and the influence of fissure permeability on the seepage field. The mixed FVM results revealed that the temperature and water pressure distributions near the fissure were closely related to the directionality of thermal conductivity in the rock mass, as well as the thermal conductivity and permeability coefficient, respectively, of the fissure itself. Comparison with results from the finite element software ABAQUS demonstrated significant advantages of the proposed method when solving temperature and seepage problems in discontinuous geological bodies containing hiatuses, mutations, and fissures.

Existence of Dissipative Solutions to the Compressible Navier-Stokes System with Potential Temperature Transport

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a strong solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.

Law of Nuclide Migration in Clayey Rocks considering Diffusion and Fluid Transport

A core concern in the research on deep geological disposal of high-level radioactive waste is the migration of radionuclides in geological bodies. Most studies on radionuclide migration consider the role of only the rock fissures without incorporating the influence of the rock matrix. In this paper, the rock mass for geological disposal of high-level radioactive waste is regarded as a fissure-pore medium. Considering the influences of radionuclide diffusion and fluid transport on radionuclide migration in the process of disposal, the governing equation of radionuclide migration and evolution in the pore-fissure medium is established. The numerical scheme of the governing equation is given based on the mixed finite volume method (FVM), using our program solution module written in C++. On this basis, the numerical test model with fissures was developed, which analyzed the radionuclide migration law in clayey rocks under various fissure and rock matrix diffusion coefficients and hydraulic conductivities. The simulation results are compared with finite element method results, revealing the superiority of the mixed FVM method in solving problems of radionuclide migration in discontinuous geological bodies containing hiatuses, mutations, and fissures. The study provides a theoretical basis for evaluating the safety, feasibility, and suitability of geological disposal repositories for high-level radioactive waste in terms of radionuclide migration.

“Mixed” Meshless Time-Domain Adaptive Algorithm for Solving Elasto-Dynamics Equations

A time-domain adaptive algorithm was developed for solving elasto-dynamics problems through a mixed meshless local Petrov-Galerkin finite volume method (MLPG5). In this time-adaptive algorithm, each time-dependent variable is interpolated by a time series function of n-order, which is determined by a criterion in each step. The high-order series of expanded variables bring high accuracy in the time domain, especially for the elasto-dynamic equations, which are second-order PDE in the time domain. In the present mixed MLPG5 dynamic formulation, the strains are interpolated independently, as are displacements in the local weak form, which eliminates the expensive differential of the shape function. In the traditional MLPG5, both shape function and its derivative for each node need to be calculated. By taking the Heaviside function as the test function, the local domain integration of stiffness matrix is avoided. Several numerical examples, including the comparison of our method, the MLPG5–Newmark method and FEM (ANSYS) are given to demonstrate the advantages of the presented method: (1) a large time step can be used in solving a elasto-dynamics problem; (2) computational efficiency and accuracy are improved in both space and time; (3) smaller support sizes can be used in the mixed MLPG5.

Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers

We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed.

A Near-Shore Linear Wave Model with the Mixed Finite Volume and Finite Difference Unstructured Mesh Method

The near-shore and estuary environment is characterized by complex natural processes. A prominent feature is the wind-generated waves, which transfer energy and lead to various phenomena not observed where the hydrodynamics is dictated only by currents. Over the past several decades, numerical models have been developed to predict the wave and current state and their interactions. Most models, however, have relied on the two-model approach in which the wave model is developed independently of the current model and the two are coupled together through a separate steering module. In this study, a new wave model is developed and embedded in an existing two-dimensional (2D) depth-integrated current model, SRH-2D. The work leads to a new wave–current model based on the one-model approach. The physical processes of the new wave model are based on the latest third-generation formulation in which the spectral wave action balance equation is solved so that the spectrum shape is not pre-imposed and the non-linear effects are not parameterized. New contributions of the present study lie primarily in the numerical method adopted, which include: (a) a new operator-splitting method that allows an implicit solution of the wave action equation in the geographical space; (b) mixed finite volume and finite difference method; (c) unstructured polygonal mesh in the geographical space; and (d) a single mesh for both the wave and current models that paves the way for the use of the one-model approach. An advantage of the present model is that the propagation of waves from deep water to shallow water in near-shore and the interaction between waves and river inflows may be carried out seamlessly. Tedious interpolations and the so-called multi-model steering operation adopted by many existing models are avoided. As a result, the underlying interpolation errors and information loss due to matching between two meshes are avoided, leading to an increased computational efficiency and accuracy. The new wave model is developed and verified using a number of cases. The verified near-shore wave processes include wave shoaling, refraction, wave breaking and diffraction. The predicted model results compare well with the analytical solution or measured data for all cases.

A priori and a posteriori estimates of stabilized mixed finite volume methods for the incompressible flow arising in arteriosclerosis

The flow arising in arteriosclerosis is modeled by the incompressible flow with a slip boundary condition of friction type in this paper, whose weak solution satisfies a variational inequality. We develop and analyze the stabilized mixed finite volume methods for the resulting model. We first achieve the same optimal order results of the finite volume methods as those of the corresponding finite element methods under the same regularity on the exact solution and a slightly additional regularity on the source term. Also, a super-close result between the respective solutions of the finite element methods and finite volume methods is derived for the first time. For adaptivity, we further derive a posterior estimate for the present problem satisfying the variational inequality, which reduces the computational cost substantially and provides efficient error control for the solution. Finally, numerical results are shown to support the developed convergence theory.

A posteriori error estimates for mixed finite volume solution of elliptic boundary value problems

Abstract The major emphasis of this work is the derivation of a posteriori error estimates for the mixed finite volume discretization of second-order elliptic equations. The estimates are established for meshes consisting of simplices on unstructured grids. We consider diffusion problems with nonhomogeneous diffusion coefficients. The error estimates are of residual types and are formulated in the energy semi-norm for a locally postprocessed approximate solutions. The estimates are fully computable and locally efficient that they can serve as indicators for adaptive refinement and for the actual control of the error. Numerical results are shown for two test examples in two space dimensions. It is found that the proposed adaptive mixed finite volume method offers a robust and accurate approach for solving second-order elliptic equations, even when highly nonhomogeneous diffusion coefficients are used in the simulations.

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018

Mixed finite volume method for linear thermoelasticity at all Poisson’s ratios

ABSTRACTA mixed (displacement/pressure) method for modeling both compressible and incompressible linear thermoelastic body problems is described. The method uses a form of Hooke’s law that introduces the pressure as an additional dependent variable and removes the numerical problems associated with incompressibility. A cell-centerd finite volume discretization by arbitrary polyhedral cells, an implicit temporal discretization scheme, both second-order accurate and segregated solution procedure make the method computationally efficient.The results of calculations are compared with analytical or finite element solutions showing a very good agreement. The present method is also compared with the displacement finite volume method and shown that it is accurate and stable for all Poisson’s ratios, while the performance of the displacement method rapidly deteriorates as the Poisson’s ratio approaches the incompressibility limit.

A Mortar Mixed Finite Volume Method for Elliptic Problems on Non-matching Multi-block Triangular Grids

A mixed finite volume method is considered for the mixed formulation of second-order elliptic equations. The computational domain can be decomposed into non-overlapping sub-domains or blocks and the diffusion tensors may be discontinuous across the sub-domain boundaries. We define a conforming triangular partition on each sub-domain independently, and employ the standard mixed finite volume method within each sub-domain. A mortar finite element space is introduced to approximate the trace of the pressure on the non-matching interfaces. Moreover, a continuity condition of flux is imposed weakly. We prove the scheme’s first order optimal rate of convergence for both the pressure and the velocity. Numerical experiments are provided to illustrate the error behavior of the scheme and confirm our theoretical results.

An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid

We develop a numerical scheme for a two-phase immiscible flow in heterogeneous porous media using a structured grid finite element method, which has been successfully used for the computation of various physical applications involving elliptic interface equations (Li et al., 2003, 2004; Chou et al., 2010; Kwak et al., 2010; Chang and Kwak, 2011). The proposed method is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use an IFEM based on the Rannacher–Turek Rannacher and Turek (1992) nonconforming space, which is a modification of the work in Kwak et al. (2010) where ‘broken’ P1 nonconforming element of Crouzeix–Raviart (1973) [1] was developed.For the Darcy velocity, we apply the mixed finite volume method studied in Chou et al. (2003) and Kwak et al. (2010) on the basis of immersed finite element method (IFEM). In this way, the Darcy velocity of the flow can be computed cheaply (locally) after we solve the pressure equation. The computed Darcy velocity is used to solve the saturation equation explicitly. The whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media. In fact, we apply a new version of multigrid algorithm to solve the pressure equation for which the CPU time grows like O(N). Numerical tests for analytic problems show that our method is almost optimal in all variables. Problem with sources/sinks is also computed and it seems to work well.

Uniform temporal convergence of numerical schemes for miscible flow through porous media

The Hybrid Mimetic Mixed (HMM) family of schemes contains the Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods. This note proves that HMM schemes, when applied to a model of the miscible displacement of one incompressible fluid by another through a porous medium, produce approximations of the concentration variable that converge uniformly towards the exact concentration.

Adaptive stabilized mixed finite volume methods for the incompressible flow

In this article, we study adaptive stabilized mixed finite volume methods for the incompressible flows approximated using the lower order elements. A residual type of a posteriori error estimator is designed and studied with the derivation of upper and lower bounds between the exact solution and the finite volume solution. A discrete local lower bound between two successive finite volume solutions is also obtained. Also, convergence of the adaptive stabilized mixed finite volume methods is established. The presented methods have three prominent features. First, it is of practical convenience in real applications with the same partitions for velocity and pressure. Second, less computational time is required by easily applying both the lower order elements and the local grid refinement necessary for the elements of interest. Third, compared with the standard finite element method, its analysis of H1-norm and L2-norm for the velocity and pressure are usually derived without any high order regularity conditions on the exact solution. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1424–1443, 2015

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

Diffusion processes have traditionally been modeled using the classical parabolic advection‐diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non‐Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro‐differential models (IDMs) have been also proposed to describe non‐Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection‐diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two‐dimensional IDM composed by a integro‐differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh. Copyright © 2015 John Wiley & Sons, Ltd.

A Posteriori Error Estimates for Mixed Finite Element and Finite Volume Methods for Parabolic Problems Coupled through a Boundary

We examine two discretization schemes for solving a pair of parabolic problems with significantly different spatial and temporal scales that are coupled through a common interface: a mixed finite element method which uses a rigorous mortar element technique in both space and time for coupling and a finite volume method which employs popular ad hoc projections for coupling. We derive a posteriori error estimates to quantify the distinct sources of error for both methods as well as discuss implementation of the estimates. The estimates include expressions that quantify the effects of using a finite number of iterations when solving the discrete equations yielding the approximations. We use the estimates to investigate the consequences of particular discretization choices on the accuracy of the computations.

A posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with nonmatching grids

Abstract : The primary purpose of this paper is to compare the accuracy and performance of two numerical approaches to solving systems of partial differential equations. These equations are posed on adjoining domains sharing boundary conditions on a common boundary interface in the important case when the meshes used on the two domains are non-matching across the interface. The first widely used approach is based on a finite volume method employing ad hoc projections to relate approximations on the two domains across the interface. The second approach uses the mathematically-founded mortar mixed finite element method. To quantify the performance, we use a goal-oriented a-posteriori error estimate that quantifies various aspects of discretization error to the overall error. While the performance difference may be not a surprise in some cases, we believe that there is a perception in part of the scientific community concerned with multiphysics systems that if the solution is smooth near the interface, then there is little effect from varying the coupling technique. We find that, on the contrary, the error associated with ad hoc coupling approaches may be large in practical situations. Moreover, we also show that mortar methods can be used with black box component solves, thus permitting an efficient and practical implementation also within legacy codes.