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Zbl 0927.65107
Dawson, Clint N.; Wheeler, Mary F.; Woodward, Carol S.
A two-grid finite difference scheme for nonlinear parabolic equations.
(English)
[J] SIAM J. Numer. Anal. 35, No.2, 435-452 (1998). ISSN 0036-1429; ISSN 1095-7170/e

The authors consider the nonlinear parabolic equation $${\partial p\over\partial t}- \nabla\cdot(K(x,p)\nabla p)= f(t,x)\quad\text{in }(0, T]\times \Omega$$ with the initial condition $p(0,x)= p^0(x)$ in $\Omega$ and Neumann boundary condition $(K(x,p)\nabla p)\cdot\nu= g$ on $(0,T]\times \Gamma$, where $\Omega$ is a rectangular domain in $\bbfR^d$ $(1\le d\le 3)$, $\Gamma$ is its boundary, $\nu$ is the outward unit normal vector on $\Gamma$ and $K:\Omega\times \bbfR\to\bbfR^{d\times d}$ is a symmetric, positive definite second-order diagonal tensor.\par Using the variational formulation of the above problem they obtain a two-level nonlinear cell-centered finite difference scheme on a coarse'' grid of mesh size $H$. For $f$ belonging to $L^2(\Omega)$ on each time-level and for $K$ being of class $C^1$ the uniqueness and the existence of the solution to the discrete problem is proved, when $\Delta t$ is small enough, where $\Delta t= \max_n \Delta t^n$, $\Delta t^n= t^{n+1}- t^n$. An a priori error estimation of order $O(H^2+\Delta t)$ is also given. (Here $d=2$; in general, the order of the error estimation is $O(H^{4-d| 2}+ \Delta t)$. Next, the authors construct a linear difference scheme on a fine'' grid of mesh size $h$, $h\ll H$ using the nonlinear solution on the coarse grid. The a priori error estimation is now of order $O(H^{4-d| 2}+ h^2+\Delta t)$.\par No computational results are given, but they are announced to appear in later papers.
[S.Burys (Kraków)]
MSC 2000:
*65M06 Finite difference methods (IVP of PDE)
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)
65M55 Multigrid methods; domain decomposition (IVP of PDE)

Keywords: error estimates; finite differences; convergence; multigrid method; two-grid method; nonlinear parabolic equation

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