Some least-squares Galerkin procedures for first-order time-dependent convection–diffusion system

Abstract

Four least-squares Galerkin finite element schemes are formulated to solve first-order time-dependent convection–diffusion systems and the convergence of these schemes is analyzed.

Introduction

The purpose of this paper is to analyze the least-squares Galerkin finite element methods for solving first-order time-dependent convection–diffusion systems. Let $\text{Ω}$ be an open bounded domain $\text{R}{}^{\mathrm{d}}$, 1⩽d⩽3, with a Lipschitz continuous boundary Γ. We consider the following initial-boundary value problem of a first-order time-dependent convection–diffusion system

$$\text{c(x)}\text{∂}\text{u(x,t)}\text{∂}\text{t}\text{+}\text{div}\text{σ(x,t)+q(x,t)u(x,t)=f(x,t),}\text{x∈}\text{Ω}\text{,}\text{0<t⩽T,}$$$$\text{σ(x,t)+A(x)∇u(x,t)+}\text{b}\text{(x,t)u(x,t)=0,}\text{x∈}\text{Ω}\text{,}\text{0<t⩽T,}$$$$\text{u(x,t)=0,}\text{x∈}\text{Γ}{}_{\mathrm{D}}\text{,}\text{σ(x,t)·ν(x)=0,}\text{x∈}\text{Γ}{}_{\mathrm{N}}\text{,}\text{0⩽t⩽T,}$$$$\text{u(x,0)=u}{}_0\text{(x),}\text{x∈}\text{Ω}\text{,}$$

where div is the divergence operator and ∇ the gradient operator, flow field $\text{b}\text{=(b}{}_1\text{(x,t),…,b}{}_{\mathrm{d}}\text{(x,t))}{}^{\mathrm{<mtext>T</mtext>}}$, source term q=q(x,t)⩾0 and exterior flow function f=f(x,t) in (1.1) are some given functions, the coefficient c=c(x) is a positive function and the diffusion coefficient matrix A=(a_ij(x))_d×d is a symmetric uniformly positive definite matrix, i.e., there exist some positive constants $\text{c}{}_{\mathrm{\ast }}$ and $\text{a}{}_{\mathrm{\ast }}$ such that

$$\text{a}{}_{\mathrm{\ast }}\text{∑}\text{i=1}\text{d}\text{ξ}{}^2{}_{\mathrm{i}}\text{⩽}\text{∑}\text{i,j=1}\text{d}\text{a}{}_{\mathrm{ij}}\text{(x)ξ}{}_{\mathrm{i}}\text{ξ}{}_{\mathrm{j}}\text{,}\text{c}{}_{\mathrm{\ast }}\text{⩽c(x),}\text{∀ξ∈}\text{R}{}^{\mathrm{d}}\text{,}\text{x∈Ω,}$$

and Γ=Γ_D∪Γ_N and ν is the unit vector normal to Γ_N.

Much research on least-squares finite element schemes and their applications to various boundary value problems of elliptic equations or systems has been done and some systematic theory on the ellipticity of schemes and convergence of approximate solutions have also been established, e.g., see Refs. [2], [6], [7], [8], [9], [10], [11], [15], [17], [21], [22]. In recent years, the least-squares finite element methods have been extended to unstationary problems, e.g., see Refs. [12], [14], [16], [20], [24], in which some very effective numerical results were shown. However, the theory on the convergence of least-squares finite element methods for evolutionary problems has not been clarified.

Our main results in this paper are that four least-squares Galerkin finite element procedures are formulated to solve the first-order time-dependent convection–diffusion problems (1.1) in Section 2 and the convergence theory on these algorithms is established in Section 3.