Abstract
We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).
Highlights
Let Ω ⊂ R2 be a bounded convex domain with smooth boundary Γ, and consider the initial-boundary value problem for the following second-order nonlinear hyperbolic equation utt − ∇ ⋅ (A (u) ∇u) = f (x, t), x ∈ Ω, 0 < t ≤ T, u (x, t) = 0, x ∈ Γ, 0 < t ≤ T, (1)u (x, 0) = u0 (x), ut (x, 0) = u1 (x), x ∈ Ω, where utt and ut denote ∂2u/∂t2 and ∂u/∂t, respectively. x = (x1, x2)
To solve problem (1), we introduce two-grid algorithms into finite element method
This method involves a nonlinear solution on the coarse grid space and a linear solution on the fine grid space
Summary
The finite element analysis for the second-order linear hyperbolic equations was discussed by Dupont [15] and Baker [16] They have obtained optimal L∞(L2) estimates for the error, O(hr), using subspaces of piecewise polynomial functions of degree ≤ r − 1, for r ≥ 1. For second-order hyperbolic equations with a nonlinear reaction term, Chen and Liu [14] have presented a two-grid method using finite volume element method and obtained error estimate in the H1-norm. In this paper, based on two conforming piecewise linear finite element spaces SH and Sh on one coarse grid with grid size H and one fine grid with grid size h, respectively, we consider the two-grid finite element discretization techniques for the second-order nonlinear hyperbolic problems. Throughout this paper, the letter C or with its subscript denotes a generic positive constant which does not depend on the mesh parameters and may be different at its different occurrences
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