Two-grid scheme for characteristics finite-element solution of nonlinear convection diffusion problems☆
Introduction
Convection–diffusion equations occur in numerous applications, noteworthy among them being mathematical models of environment pollution and neutron transportation. In this paper we discuss the nonlinear formwe use the modified method of characteristics finite-element developed by Douglas–Russell [1], to discrete in space with equal-order accuracy in u. To linearize the resulting discrete equations, we use a two-grid scheme, which allows us to iterate on a grid much coarser than that used for the final solution.
We owe the impetus for using a two-grid approach to Xu [2], [3], [8], who demonstrates the possibility of mapping a fine-grid Galerkin finite-element problem onto a coarse grid. In that context, one solves the nonlinear problem via Newton-like iterations. After convergence on the coarse grid, one then extrapolates back to the fine grid using a Taylor expansion. Dawson and Wheeler [4] extend the method to nonlinear reaction–diffusion equations. Li Wu and Myron [5] extend it to another nonlinear reaction–diffusion equations.
The remainder of the paper is organized as follows: Section 2 describes the algorithm. Section 3 gives an error analysis establishing the method’s convergence. Section 4 presents some computational results confirming the method’s utility, and Section 5 draws conclusions.
We assume that the coefficients a, b and c are bounded and that
- (H1)
- (H2)
Throughout this paper, we shall use the letter C to denote a generic positive constant which may for different values at its different occurrences. Let Wm,p(Ω) denote the Sobolev space on I. And the norm is defined byWe write Wm,2(Ω) = Hm(Ω), ∥v∥Hm(Ω) = ∥v∥m, and ∥v∥L2(Ω) = ∥v∥0 = ∥v∥ and also assume that the solution u of (1.1) satisfies:
- (H3)
u ∈ L∞(0,T;Hr + 1(I)),
- (H4)
if s = 1 and r ⩾ 2 if s = 0,
- (H5)
.
Section snippets
Characteristics finite-element discretization
We begin by briefly reviewing the characteristics finite-element discretization of the problem (1.1). Letand let the characteristic direction associated with the operator cut + b ux be denoted by τ = τ(x,t), whereThen, Eqs. (1.1) can be put in form
We introduce the Sobolev space , and the notation (u,v) = ∫Iu(x)v(x)dx,∀u,v∈V. Let . Then, multiplying the
Convergence analysis
The two-grid method affords a remarkably efficient linearization. It is possible to execute all of the Newton-like iterations on very coarse grids, then use the heuristic (2.9) to obtain an accurate fine-grid solution in one additional step. We devote the rest of this section to the analysis of the two-grid scheme. We begin by describing the discretization space and associated projection operator. We then give a convergence analysis for the characteristics finite-element method, using the key
Numerical example
To illustrate the effectiveness of the two-grid linearization, we examine the following simple test problem:where a(x) = 0.001. The G(x,t) is determined by the exact solution u(x,t) = (1 − x)ext.
We solve (4.1) by Δt = 0.125 × 10−5 from t = 0 to 0.20. For H = 2−4, h = H7/4 = 2−7, we use the two-grid scheme presented in this paper. First, we give the coarse-grid partition ΔH with H = 2−4 (Fig. 2) and get the nonlinear
Conclusion
Two-grid linearization offers an attractive way to solve the nonlinear problems involving convection–diffusion equations. The key feature of the two-grid method is that it allows one to execute all of the nonlinear iterations on a system associated with a coarse spatial grid, without sacrificing the order of accuracy of the fine-grid solution. The two-grid scheme combined with the characteristics finite-element method, cannot only decrease the numerical oscillation caused by dominated
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This work supported by National Science Foundation of China grants 10371096.