The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations

The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic intego-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L ^∞(H ¹)L ^∞(L ²) norms and quasi-optimal estimate in L ^∞(L ^∞) norm using energy arguments. Further, optimal L(L ²)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature.     

A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems

We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Biquadratic finite volume element method based on optimal stress points for second order hyperbolic equations

Based on optimal stress points, we develop a full discrete finite volume element scheme for second order hyperbolic equations using the biquadratic elements. The optimal order error estimates in L^∞(H¹), L^∞(L²) norms are derived, in addition, the superconvergence of numerical gradients at optimal stress points is also discussed. Numerical results confirm the theoretical order of convergence. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Semidiscrete Galerkin method for equations of motion arising in Kelvin‐Voigt model of viscoelastic fluid flow

Finite element Galerkin method is applied to equations of motion arising in the Kelvin–Voigt model of viscoelastic fluids for spatial discretization. Some new a priori bounds which reflect the exponential decay property are obtained for the exact solution. For optimal L^∞( L ²) estimate in the velocity, a new auxiliary operator which is based on a modification of the Stokes operator is introduced and analyzed. Finally, optimal error bounds for the velocity in L^∞( L ²) as well as in L^∞( H )‐norms and the pressure in L^∞(L²)‐norm are derived which again preserves the exponential decay property. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations

A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi‐discrete and a family of fully‐discrete time approximate schemes are formulated. These schemes are symmetric. Hp‐version error estimates are analyzed for these schemes. For the semi‐discrete time scheme a priori L^∞(H¹) error estimate is derived and similarly, l^∞(H¹) and l²(H¹) for the fully‐discrete time schemes. These results indicate that spatial rates in H¹ and time truncation errors in L² are optimal. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

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

The 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     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

Solution operator approximations for characteristic roots of delay differential equations

In this paper a new method for the numerical computation of characteristic roots for linear autonomous systems of Delay Differential Equations (DDEs) is proposed. The new approach enlarges the class of methods recently developed (see [SIAM J. Numer. Anal. 40 (2002) 629; D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: experimental comparison, in: BIOCOMP2002: Topics in Biomathematics and Related Computational Problems at the Beginning of the Third Millennium, Vietri, Italy, 2002, Sci. Math. Jpn. 58 (2) pp. 377–388; D. Breda, The infinitesimal generator approach for the computation of characteristic roots for delay differential equations using BDF methods, Research Report RR2/2002, Department of Mathematics and Computer Science, Università di Udine, Italy, 2002; IMA J. Numer. Anal. 24 (2004) 1; SIAM J. Sci. Comput. (2004), in press]) and in particular it is based on a Runge–Kutta (RK) time discretization of the solution operator associated with the system. Hence this paper revisits the Linear Multistep (LMS) approach presented in [SIAM J. Numer. Anal. 40 (2002) 629] for the multiple discrete delay case and moreover extends it to the distributed delay case. We prove that the method converges with the same order as the underlying RK scheme and illustrate this with some numerical tests that are also used to compare the method with other existing techniques.     

Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations

In this article, we investigate interior penalty discontinuous Galerkin (IPDG) methods for solving a class of two‐dimensional nonlinear parabolic equations. For semi‐discrete IPDG schemes on a quasi‐uniform family of meshes, we obtain a priori bounds on solutions measured in the L² norm and in the broken Sobolev norm. The fully discrete IPDG schemes considered are based on the approximation by forward Euler difference in time and broken Sobolev space. Under a restriction related to the mesh size and time step, an hp ‐version of an a priori l^∞(L²) and l²(H¹) error estimate is derived and numerical experiments are presented.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 288–311, 2012     

A handy proof of Gay's theorem

In 1979, Gay proved that Broyden's methods, when used for n‐square linear systems, terminate in at most 2n iterations (SIAM J. Numer. Anal. 1979; 16 :623–630). Also, the ABS methods were introduced in 1984 (Numer. Math. 1984; 45 :1361–1376). In this paper we show another (handy) proof of Gay's theorem by these algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

On the finite element approximation of elliptic QVIs with noncoercive operators

In this paper, we extend the approach developed by the author for the standard finite element method in the L^∞‐norm of the noncoercive variational inequalities (VI) (Numer Funct Anal Optim.2015;36:1107‐1121.) to impulse control quasi‐variational inequality (QVI). We derive the optimal error estimate, combining the so‐called Bensoussan‐Lions Algorithm and the concept of subsolutions for VIs.     

On some subclasses of P‐matrices

A matrix with positive row sums and all its off‐diagonal elements bounded above by their corresponding row averages is called a B‐matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22 :1027–1037) and (Numer. Math. 2003; 95 :337–345). In this paper, it is generalized to more extended matrices—MB‐matrices, which is proved to be a subclass of the class of P‐matrices. Subsequently, we establish relationships between defined and some already known subclasses of P‐matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21 :67–78; Linear Algebra Appl. 2004; 393 :353–364; Linear Algebra Appl. 1995; 225 :117–123). As an application, some subclasses of P‐matrices are used to localize the real eigenvalues of a real matrix. Copyright © 2007 John Wiley & Sons, Ltd.     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010