An Eulerian‐Lagrangian substructuring domain decomposition method for unsteady‐state advection‐diffusion equations

We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001     

An Eulerian‐Lagrangian discontinuous Galerkin method for transient advection‐diffusion equations

We develop an Eulerian‐Lagrangian discontinuous Galerkin method for time‐dependent advection‐diffusion equations. The derived scheme has combined advantages of Eulerian‐Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem‐dependent parameter. An optimal‐order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Accuracy of multivalue methods during change of time‐step size: Adams‐Moulton

Multivalue methods are a class of time‐stepping procedures for numerical solution of differential equations that progress to a new time level using the approximate solution for the function of interest and its derivatives at a single time level. The methods differ from multistep procedures, which make use of solutions to the differential equation at multiple time levels to advance to the new time level. Multistep methods are difficult to employ when a change in time‐step is desired, because the standard formulas (e.g., Adams‐Moulton or Gear) must be modified to accommodate the change. Multivalue methods seem to possess the desirable feature that the time‐step may be changed arbitrarily as one proceeds, since the solution proceeds from a single time level. However, in practice, changes in the time‐step introduce lower order errors or alter the coefficient in the truncation error term. Here, the multivalue Adams‐Moulton method is presented based on a general interpolation procedure. Modifications required to retain the high‐order accuracy of these methods during a change in time‐step are developed. Additionally, a formula for the unknown initial derivatives is presented. Finally, two examples are provided to illustrate the potential merit of the modification to the standard multivalue methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partials Differential Eq 16: 312–326, 2000     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Extension of high‐order compact schemes to time‐dependent problems

In this article, we present an extension of our previous approaches for steady‐state higher‐order compact (HOC) difference methods to time‐dependent problems. The formulation also provides a framework for similar treatment of other HOC spatial schemes. A stability analysis is provided for transient convection‐diffusion in 1D and transient diffusion in 2D. Supporting numerical experiments are included to illustrate stability and accuracy as well as oscillatory and dissipative behavior. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 657–672, 2001     

A single‐node characteristic collocation method for unsteady‐state convection‐diffusion equations in three‐dimensional spaces

We develop a nonconventional single‐node characteristic collocation method with piecewise‐cubic Hermite polynomials for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The reduction of number of nodes has great potential for problems defined on high space dimensions, which appears in such problems as quantification of uncertainties in subsurface porous media. The method developed here is easy to formulate. Numerical experiments are presented to show the strong potential of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 786–802, 2011     

A robust algebraic multilevel preconditioner for non‐symmetric M‐matrices

Stable finite difference approximations of convection‐diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M‐matrix, which is highly non‐symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non‐symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized’ version of the two‐level preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies for some real constant c such that . This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two‐ and multi‐level schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non‐uniform (stretched) grids. Copyright © 2000 John Wiley & Sons, Ltd.     

A nonconventional Eulerian‐Lagrangian single‐node collocation method with Hermite polynomials for unsteady‐state advection‐diffusion equations

We developed a nonconventional Eulerian‐Lagrangian single‐node collocation method (ELSCM) with piecewise‐cubic Hermite polynomials as basis functions for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The method is relatively easy to formulate. Numerical experiments are presented to show the strong potential of this method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 271–283, 2003.     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion problems

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion equations are presented and analyzed. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. The linear part of the equation is discretized implicitly and the nonlinear part of the equation explicitly. The schemes are stable and very efficient. We derive optimal order error estimates. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:93–104, 2001     

An Eulerian‐Lagrangian control volume scheme for two‐dimensional unsteady advection‐diffusion problems

We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Two‐grid methods for mixed finite‐element solution of coupled reaction‐diffusion systems

We develop 2‐grid schemes for solving nonlinear reaction‐diffusion systems: where p = (p, q) is an unknown vector‐valued function. The schemes use discretizations based on a mixed finite‐element method. The 2‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction‐diffusion equations. An application to prepattern formation in mathematical biology illustrates the method's effectiveness. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 589–604, 1999     

The restarted shift‐and‐invert Krylov method for matrix functions

In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift‐and‐invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag–Leffler functions are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

A combined BDF‐semismooth Newton approach for time‐dependent Bingham flow

This article is devoted to the numerical simulation of time‐dependent convective Bingham flow in cavities. Motivated by a primal‐dual regularization of the stationary model, a family of regularized time‐dependent problems is introduced. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully discrete approach is studied. A stable finite element approximation in space together with a second‐order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is used. We present two numerical experiments to verify the main properties of the proposed approach. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Strengthened Cauchy–Bunyakowski–Schwarz inequality for a three‐dimensional elasticity system

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three‐dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large‐scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real‐life elasticity problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Least‐squares spectral collocation method for the Stokes equations

First‐order system least‐squares spectral collocation methods are presented for the Stokes equations by adopting the first‐order system and modifying the least‐squares functionals in 2 . Then homogeneous Legendre and Chebyshev (continuous and discrete) functionals are shown to be elliptic and continuous with respect to appropriate product weighted norms. The spectral convergence is analyzed for the proposed methods with some numerical experiments. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 128–139, 2004     

A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the Eulerian‐Lagrangian discontinuous Galerkin method for unsteady‐state advection–diffusion equations with general inflow and outflow boundary conditions. It is well‐known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.