Alternating Direction Galerkin Method Research Articles

An alternating direction Galerkin method for a time-fractional partial differential equation with damping in two space dimensions

In this paper, we propose an efficient alternating direction implicit (ADI) Galerkin method for solving the time-fractional partial differential equation with damping, where the fractional derivative is in the sense of Caputo with order in (1,2). The presented numerical scheme is based on the L2-1_{sigma} method in time and the Galerkin finite element method in space. The unconditional stability and convergence of the numerical scheme are both carefully proved. Numerical results are displayed for supporting the theoretical analysis.

An alternating direction Galerkin method for elliptic problems with gradient nonlinearity

AbstractSeveral problems in many applications involve the solution of partial differential equations with gradient dependent nonlinearity. The numerical solution of the resulting nonlinear system is rather expensive. We present an alternating direction Galerkin method that allows much faster solution of the nonlinear system. The alternating direction formulation help reduce the problem into a sequence of nonlinear systems that may be solved very efficiently. Theoretical study of the convergence of the method will also be presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Alternating Direction Galerkin Method for a Class of Second-Order Hyperbolic Equations in Two Space Variables

A new alternating-direction implicit (ADI) Galerkin method is devised and analyzed for solving a certain class of second-order hyperbolic initial-boundary value problems in two space variables. This class includes the wave equation in Cartesian coordinates, polar coordinates, and cylindrical coordinates with radial symmetry. Optimal a priori $H_0^1 $ and $L^2 $-error estimates are derived.

Alternating direction Galerkin method of solving the third boundary value problem for two-dimensional parabolic equations

Article Alternating direction Galerkin method of solving the third boundary value problem for two-dimensional parabolic equations was published on January 1, 1987 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 2, issue 6).

Alternating direction Galerkin method with capacitance matrix for the parabolic problem with natural boundary condition

Alternating direction Galerkin method with capacitance matrix for the parabolic problem with natural boundary condition

An Extension of the Alternating Direction Galerkin Method to More General Geometries

In this paper an extension of the alternating direction Galerkin method to rectangular polygons is presented. Only Dirichlet problems with tensor product basis of piecewise linear Hermite polynomials are considered. The approach proposed is based on imbedding the rectangular polygon into a rectangle, and then using a result due to F. Stenger [5] to provide the solution of the associated algebraic problem in terms of a sequence of simpler algebraic problems. This approach has strong similarities with that proposed by J. Dendy and G. Fairweather [SIAM J. Numer. Anal., 12 (1975), pp. 144–163]; however, it is conceptually different. Furthermore, the solution is given in an algorithmic manner that is well suited to computer implementation, and it is also more general for the class of problems considered.

Alternating direction Galerkin method for quasilinear parabolic equations

Consider the following parabolic equation: (1) ∂u/∂t−∑2i=1(d/dxi)ai(x,t,u,D1u,D2u)+a0(x,t,u,D1u,D2u)=f(x,t), x=(x1,x2)∈Ω⊂R2, t∈[0,T], with the initial value condition u(x,0)=u0(x), x∈Ω, and with the boundary value condition u(x,t)=0, x∈∂Ω, t∈[0,T]. For the solution of equation (1) the author proposes a variational-difference method. Namely, he approximates equation (1) by Galerkin's method with respect to the variables x1,x2 and by the finite-difference method with respect to the variable t. Under some assumptions concerning the coefficients ai, i=0,1,2, an estimate of the error is given.

Alternating Direction Galerkin Methods for the Systems of Hyperbolic Equations of First Order

In this paper alternating direction Galerkin methods are extended to the system of hyperbolic equations of first order and to the system of parabolic-hyperbolic equations with periodic initial conditions.

An Analysis of Some Galerkin Schemes for the Solution of Nonlinear Time-Dependent Problems

A priori error estimates are derived for several Galerkin procedures. For a nonlinear parabolic initial-boundary value problem, optimal rate L2-error estimates are derived for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods. For a nonlinear second order hyperbolic initial-boundary value problem, optimal rate L2-error estimates are derived for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method. 1. Introduction. In this paper we derive a priori error estimates for several Galerkin procedures. In ?3 we consider a nonlinear parabolic initial-boundary value problem, and we derive error estimates for the forward and central difference Laplace modified Galerkin methods and the forward and central difference Laplace modified alternating direction Galerkin methods of (5); these methods were formulated in (5), but in the central difference methods proofs were given only for the case in which the elliptic part was linear. In ?4 we consider a nonlinear second order hyperbolic initial-boundary value problem, and we derive error estimates for the Laplace modified Galerkin method and the Laplace modified alternating direction Galerkin method of (5); these methods are the obvious extensions of the methods of (5) to the nonlinear case. In ?2 we present certain notation, assumptions, and known results which are common to the development of ? 3 and 4. Section 5 contains some computational results.

Galerkin Methods for Vibration Problems in Two Space Variables

Some discrete Galerkin methods are formulated for approximately solving a linear fourth order parabolic partial differential equation in $\Omega \times (0,T]$, where $\Omega $ is an arbitrary bounded domain in $\mathbb{R}^2 $. Error estimates are obtained which show that these methods are second order correct in time. When $\Omega $ is a rectangle, alternating-direction Galerkin methods are discussed.