Discontinuous Finite Volume Element Research Articles

A Quadratic Discontinuous Finite Volume Element Scheme for Stokes Problems

A Quadratic Discontinuous Finite Volume Element Scheme for Stokes Problems

An adaptive discontinuous finite volume element method for the Allen-Cahn equation

An adaptive discontinuous finite volume element method for the Allen-Cahn equation

Discontinuous finite volume element method for Darcy flows in fractured porous media

This paper presents a numerical simulation of the single phase Darcy flow model in two-dimensional fractured porous media. Under some physically consistent coupling conditions, the model can be described as a reduced problem by coupling the bulk problem in porous matrix and the fracture problem in fractures. Flows are governed by the primal form of the Darcy’s equations for both the bulk and fractures. The coupled discontinuous finite volume element methods and conforming finite element method are adopted to solve the bulk problem and fracture problem, respectively. We theoretically analyze the well-posedness of the discrete problem, and derive optimal error estimates in standard L2 error and broken H1 error. Numerical experiments include not only the fractures with high permeability as the prior flow conduit, but also the fractures with low permeability as the flow barrier, which demonstrate the accuracy, flexibility and robustness of our discrete formulation for complicated networks of fractures in porous media domain.

Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model

In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method is efficient, optimally convergent, conserving the mass, convenient to implement, flexible for mesh refinement, and easy to handle complex geometries with different types of boundary conditions. We rigorously prove the mass conservation property and the discrete energy dissipation for the proposed fully discrete discontinuous finite volume element scheme. Using numerical tests, we verify the accuracy, confirm the mass conservation and the energy law, test the influence of surface tension and small density variations, and simulate the driven cavity, the Rayleigh-Taylor instability.

一维双曲守恒律方程的保极值间断有限体积元方法

一维双曲守恒律方程的保极值间断有限体积元方法

Discontinuous finite volume element method of two-dimensional unsaturated soil water movement problem

In this paper, a numerical approximation method for the two-dimensional unsaturated soil water movement problem is established by using the discontinuous finite volume method. We prove the optimal error estimate for the fully discrete format. Finally, the reliability of the method is verified by numerical experiments. This method is not only simple to calculate, but also stable and reliable.

A Crank–Nicolson discontinuous finite volume element method for a coupled non-stationary Stokes–Darcy problem

In this paper, we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a Crank–Nicolson discretization in time, and we obtain the solution at the first time step by applying a first order backward Euler method to solve the coupling of time-dependent Stokes–Darcy problem. The proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by the piecewise constant elements. Then, the unique solvability of the approximate solution for the discrete problem is derived. In addition, the optimal error estimates of the full discretization in standard L2− norm and broken H1− norm are obtained. Finally, a series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, adaptive mesh refinement, mass conservation, capability to conveniently deal with complicated geometries with different types of boundary conditions, and the applicability to the problems with realistic parameters. In particular, the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain, it encompasses both fractures with high permeability which act as the prior flow conduit, and fractures with low permeability as a flow barrier.

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

In this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed/discontinuous approximation) measured in suitable norms are derived, showing optimal orders of convergence.

Three First-Order Finite Volume Element Methods for Stokes Equations under Minimal Regularity Assumptions

Three first-order finite volume element methods, namely, conforming, nonconforming, and discontinuous Galerkin schemes for Stokes equations, are analyzed and compared using the medius analysis. The latter analysis is based on a combination of arguments from a priori and a posteriori error analyses under no extra regularity assumptions on the weak solution. Best-approximation results for the energy norms hold in all cases and allow for comparison results up to generic equivalence constants and higher-order data oscillation of the applied volume forces with little modification for different pressure approximations. A priori and a posteriori analyses of discontinuous Galerkin finite volume element methods with SIPG, IIPG, NIPG are included for completeness.

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard $$L^2$$ -norm and broken $$H^1$$ -norm are obtained for three discontinuous finite volume element methods (symmetric, non-symmetric and incomplete types). A series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, mass conservation, capability to deal with complicated geometries, and applicability to the problems with realistic parameters.

A Discontinuous Finite Volume Element Method Based on Bilinear Trial Functions

We propose a discontinuous finite volume element (DFVE) method for second order elliptic and parabolic problems. Discontinuous bilinear functions are used as the trial functions. We give the stability analysis of this DFVE method and derive the optimal error estimates in the broken [Formula: see text]-norm. Specifically, the optimal [Formula: see text]-error is obtained for the first time for the bilinear DFVE methods solving elliptic and parabolic problems.

A discontinuous method for oil‐water flow in heterogeneous porous media

AbstractAn oil‐water flow‐transport problem in a heterogeneous porous medium (HPM), given by the Brinkman model coupled with a fractional flow formulation, is solved numerically by a discontinuous finite volume element (DFVE) method coupled with a Runge‐Kutta Discontinuous Galerkin (RKDG) scheme. A new (original) numerical example is presented. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi–Douglas–Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed–primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and Boycott effects). The applicability of the method is not limited to flow in porous media, but can also be employed to describe many other physical systems governed by a similar set of equations, including e.g. multi-component materials.

Discontinuous approximation of viscous two-phase flow in heterogeneous porous media

Runge–Kutta Discontinuous Galerkin (RKDG) and Discontinuous Finite Volume Element (DFVE) methods are applied to a coupled flow-transport problem describing the immiscible displacement of a viscous incompressible fluid in a non-homogeneous porous medium. The model problem consists of nonlinear pressure–velocity equations (assuming Brinkman flow) coupled to a nonlinear hyperbolic equation governing the mass balance (saturation equation). The mass conservation properties inherent to finite volume-based methods motivate a DFVE scheme for the approximation of the Brinkman flow in combination with a RKDG method for the spatio-temporal discretization of the saturation equation. The stability of the uncoupled schemes for the flow and for the saturation equations is analyzed, and several numerical experiments illustrate the robustness of the numerical method.

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

The sedimentation–consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low Reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the Stokes problem written in terms of the mixture flow velocity and the pressure field. Here both the viscosity and the forcing term depend on the local solids concentration. A semi-discrete discontinuous finite volume element (DFVE) scheme is proposed for this model. The numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field, whereas the pressure is approximated by piecewise constant elements. The unique solvability of both the nonlinear continuous problem and the semi-discrete DFVE scheme is discussed, and optimal convergence estimates in several spatial norms are derived. Properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples, including flows under gravity with changing direction, a secondary settling tank in an axisymmetric setting, and batch sedimentation in a tilted cylindrical vessel.

Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem

The aim of this paper is to develop and analyze a family of stabilized discontinuous finite volume element methods for the Stokes equations in two and three spatial dimensions. The proposed scheme is constructed using a baseline finite element approximation of velocity and pressure by discontinuous piecewise linear elements, where an interior penalty stabilization is applied. A priori error estimates are derived for the velocity and pressure in the energy norm, and convergence rates are predicted for velocity in the $$L^2$$L2-norm under the assumption that the source term is locally in $$ H^1$$H1. Several numerical experiments in two and three spatial dimensions are presented to validate our theoretical findings.

A discontinuous finite volume element method for second‐order elliptic problems

AbstractIn this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L2 ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

AbstractThe incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L∞(L2) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems

AbstractIn this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L2 and broken H1‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Discontinuous finite volume element method for parabolic problems

Discontinuous finite volume element method for parabolic problems