Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

Newton method for reactive solute transport with equilibrium sorption in porous media

We present a mass conservative numerical scheme for reactive solute transport in porous media. The transport is modeled by a convection-diffusion-reaction equation, including equilibrium sorption. The scheme is based on the mixed finite element method (MFEM), more precisely the lowest-order Raviart-Thomas elements and one-step Euler implicit. The underlying fluid flow is described by the Richards equation, a possibly degenerate parabolic equation, which is also discretized by MFEM. This work is a continuation of Radu et al. (2008) and Radu et al. (2009) [1] and [2] where the algorithmic aspects of the scheme and the analysis of the discretization method are presented, respectively. Here we consider the Newton method for solving the fully discrete nonlinear systems arising on each time step after discretization. The convergence of the scheme is analyzed. In the case when the solute undergoes equilibrium sorption (of Freundlich type), the problem becomes degenerate and a regularization step is necessary. We derive sufficient conditions for the quadratic convergence of the Newton scheme.     

Numerical solutions of the space‐time fractional advection‐dispersion equation

Fractional advection‐dispersion equations are used in groundwater hydrologhy to model the transport of passive tracers carried by fluid flow in a porous medium. In this paper we present two reliable algorithms, the Adomian decomposition method and variational iteration method, to construct numerical solutions of the space‐time fractional advection‐dispersion equation in the form of a rabidly convergent series with easily computable components. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the two approaches are easy to implement and accurate when applied to space‐time fractional advection‐dispersion equations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A uniform estimate for the MMOC for two‐dimensional advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for two‐dimensional time‐dependent advection‐diffusion equations, in the sense that the generic constants in the estimates depend on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Numerical stability of the BEM for advection‐diffusion problems

A boundary element method (BEM) approach has been developed to solve the time‐dependent 1D advection‐diffusion equation. The 1D solution is part of a 3D numerical scheme for solving advection‐diffusion (AD) problems in fractured porous media. The full 3D scheme includes a 3D solution for the porous matrix, which is coupled with a 2D solution for fractures and a 1D solution for fracture intersections. As the hydraulic conductivity of the fracture intersections is usually higher than the hydraulic conductivity of the fractures and by at least one order of magnitude higher than the hydraulic conductivity of the porous matrix, the fastest flow and solute transport occurs in the fracture intersections. Therefore it is important to have an accurate and stable 1D solution of the transient AD problems. This article presents two different 1D BEM formulations for solution of the AD problems. The particular advantage of these formulations is that they provide one of the most straightforward and simplest ways to couple multiple intersecting 2D Boundary Element problems discretized with linear discontinuous elements. Both formulations are tested and compared for accuracy, stability, and consistency. The analysis helps to select the more suitable formulations according to the properties of the problem under consideration. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

An optimal‐order error estimate for MMOC and MMOCAA schemes for multidimensional advection‐reaction equations

In this article, we analyze the modified method of characteristics (MMOC) and an improved version of the MMOC, named the modified method of characteristics with adjusted advection (MMOCAA), for multidimensional advection‐reaction transport equations in a uniform manner. We derive an optimal‐order error estimate for these schemes. Numerical results are presented to verify the theoretical estimates. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 69–84, 2002     

Block‐centered upwind multistep difference method and convergence analysis for numerical simulation of oil reservoir

The three‐dimensional displacement of two‐phase flow in porous media is a preliminary problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a nonlinear system of partial differential equations to describe incompressible miscible case. The pressure is defined by an elliptic equation, and the concentration is defined by a convection‐dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block‐centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block‐centered upwind multistep method to solve the concentration, where the time derivative is approximated by multistep method, and the diffusion term and convection term are treated by a block‐centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve such a convection‐dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block‐centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well‐known model problem.     

Well‐posedness of a generalized time‐harmonic transport equation for acoustics in flow

We study a generalized time‐harmonic transport equation, which appears in the Goldstein equations and allows us to model the acoustic radiation in a flow. We investigate the well‐posedness of this transport problem. The result will be established under the assumption of a Ω‐filling flow, which, in 2D, is simply equivalent to a flow that does not vanish. The approach relies on the method of characteristics, which leads to the resolution of the transport equation along the streamlines, and on general results of functional analysis. The theoretical results are illustrated with numerical results obtained with a Streamline Upwind Petrov‐Galerkin finite element scheme.     

An optimal‐order error estimate for a Galerkin‐mixed finite‐element time‐stepping procedure for porous media flows

This article deals with the numerical approximation of miscible displacement problem of one incompressible fluid in a porous medium. The adopted formulation is based on the combined use of a mixed finite‐element scheme to treat pressure equation and of the finite‐element approach to treat concentration equation. Optimal‐order error estimates are obtained under some milder mesh‐parameter constraints. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 707–719, 2012     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

A second‐order backward difference time‐stepping scheme for penalized Navier–Stokes equations modeling filtration through porous media

We investigate the stability and convergence of a fully implicit, linearly extrapolated second‐order backward difference time‐stepping scheme for the penalized Navier–Stokes equations modeling filtration through porous media. In the penalization approach, an extended Navier–Stokes equation is used in the entire computational domain with suitable resistance terms to mimic the presence of porous medium. It is widely used as an alternative to the heterogeneous approach in which different types of partial differential equations (PDEs) are used in fluid and porous subregions along with suitable continuity conditions at the interface. However, the introduction of extra resistance terms makes the penalized Navier–Stokes equations more nonlinear. We prove that the linearly extrapolated scheme is unconditionally stable and derive optimal order error estimates without any stability condition. To show feasibility and applicability of the approach, it is used to numerically solve a passive control problem in which flow around a solid body is controlled by adding porous layers on the surface. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 681–705, 2016     

An optimal‐order error estimate on an H1‐Galerkin mixed method for a nonlinear parabolic equation in porous medium flow

We present an H¹‐Galerkin mixed finite element method for a nonlinear parabolic equation, which models a compressible fluid flow process in subsurface porous media. The method possesses the advantages of mixed finite element methods while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. We conducted theoretical analysis to study the existence and uniqueness of the numerical solutions of the scheme and prove an optimal‐order error estimate for the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

The boundary integral equation approach for numerical solution of the one‐dimensional Sine‐Gordon equation

This article describes a numerical method based on the boundary integral equation and dual reciprocity method for solving the one‐dimensional Sine‐Gordon (SG) equation. The time derivative is approximated by the time‐stepping method and a predictor–corrector scheme is employed to deal with the nonlinearity which appears in the problem. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. In addition, the conservation of energy in SG equation is investigated. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Finite element method for two‐dimensional time‐fractional tricomi‐type equations

In this article, we consider the finite element method (FEM) for two‐dimensional linear time‐fractional Tricomi‐type equations, which is obtained from the standard two‐dimensional linear Tricomi‐type equation by replacing the first‐order time derivative with a fractional derivative (of order α, with 1 <α< 2 ). The method is based on finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the error estimate is presented. The comparison of the FEM results with the exact solutions is made, and numerical experiments reveal that the FEM is very effective. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.