Two-grid scheme for characteristics finite-element solution of nonlinear convection diffusion problems

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Abstract

A two-grid scheme for characteristics finite-element solution of nonlinear convection-dominated diffusion equations was constructed. The L2 and H1 error estimates of the scheme were derived. A numerical example was also presented. The scheme involves solving one small, nonlinear problem on the coarse-grid system, one linear problem on the fine-grid system. The error estimates and the numerical example show that the new scheme is efficient to the nonlinear convection-dominated diffusion equations.

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 formc(x)ut+b(x)ux-xa(x)ux=f(u,x,t),(x,t)I×(0,T],u(x,t)=0,(x,t)I×(0,T],u(x,0)=u0(x),xI,we 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)

    0<a0a(x),0<c0c(x),|b(x)c(x)|+|ddx(b(x)c(x))|C,xI

  • (H2)

    |fu|+|2fu2|C,xI

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 byvWm,p=αmDαvLp(Ω)p1/p,1p<,maxαmesssup|Dαv|,p=.We write Wm,2(Ω) = Hm(Ω), ∥vHm(Ω) = vm, and ∥vL2(Ω) = v0 = v∥ and also assume that the solution u of (1.1) satisfies:

  • (H3)

     u  L(0,T;Hr + 1(I)),

  • (H4)

    utL2(0,T;Hr+s),r=1 if s = 1 and r  2 if s = 0,

  • (H5)

    2ut2L2(0,T;L2(I)).

Section snippets

Characteristics finite-element discretization

We begin by briefly reviewing the characteristics finite-element discretization of the problem (1.1). Letψ(x)=c2(x)+b2(x)and let the characteristic direction associated with the operator cut + b ux be denoted by τ = τ(x,t), whereτ=c(x)ψ(x)t+b(x)ψ(x)x.Then, Eqs. (1.1) can be put in formψ(x)uτ-xa(x)ux=f(u,x,t),xI,t(0,T],u(x,t)=0,xIt(0,T],u(x,0)=u0(x),xI.

We introduce the Sobolev space V=H01(I), and the notation (u,v) = Iu(x)v(x)dx,∀u,vV. Let A(u,v)=(aux,vx). 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:ut+ux-xa(x)ux=-u2+G(x,t),x[0,1],t(0,T],u(0,t)=1,u(1,t)=0,t(0,T],u(x,0)=1-x,x[0,1],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.

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