一种半隐式有限体积-有限元方法的稳定性

The Stability of a Semi-implicit Finite Volume-Finite Element Method

研究非线性对流扩散问题的一种半隐式有限体积和有限元方法相结合的离散方法 。

Let Ω∈R 2 be a bounded polygonal domain. In the space-time cylinder Q T=Ω×(0, T)(0<T<∞), we consider the following initial-boundary value problem:u t+∑2i=1 f i(u)x i-a△u=g(x, t), (x, t)∈Q T u| Ω×(0, T)=0, u(x, 0)=u 0(x), x∈Ωwhere a>0 is a given constant, f i∈C 1(R), f i(0)=0, g∈C(;W 1, q(Ω)), u 0∈W 1, p 0(Ω), for some p, q>2. We investigate a numerical method in which nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. The semi-implicit discret formulation considered in this paper for weak form of the problem is1τ(u n+1 h-u n h, v h) h+b h(u n h, v h)+a((u n+1 h+u n h2), v h)=(g n+1, v h) h, v h∈V h, t n∈[0, T).where b h(u n h, v h) is the finite volume approximate term. Under the assumption that the triangulations are of weakly acute type, we prove that the approximate solution u n h is bounded in L ∞(Q T) and L 2(0, T; H 1 0(Ω)), and for the Fourier transform [AKw^D] h τ of w h τ(w h τ is a continous, piecewise linear interpolation function of u n h), |s| γ h τ(s)(0<γ<14) is bounded in L 2(0, T; L 2(Ω)).