Abstract
A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.
Highlights
Let Ω ⊂ R2 be a bounded convex domain with smooth boundary Γ and consider the initialboundary value problem for the following nonlinear parabolic equations: ut − ∇ · A u ∇u f x, x ∈ Ω, 0 < t ≤ T, u x, t 0, x ∈ Γ, 0 < t ≤ T, 1.1 u x, 0 u0 x, x ∈ Ω, where ut denotes ∂u/∂t. x x1, x2, f x is a given real-valued function on Ω
We adopt the standard notation for Sobolev spaces Ws,p Ω with 1 ≤ p ≤ ∞ consisting of functions that have generalized derivatives of order s in the space Lp Ω
We denote by H01 Ω the subspace of H1 Ω of functions vanishing on the boundary Γ
Summary
A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. In this twogrid scheme, the full nonlinear problem is solved only on a coarse grid with grid size H. The nonlinearities are expanded about the coarse grid solution on a fine gird of size h, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the H1-norm O h H2 which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy h O H2. An example is given to illustrate the theoretical results
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