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A two-grid finite difference scheme for nonlinear parabolic equations. (English) Zbl 0927.65107

The authors consider the nonlinear parabolic equation

$\frac{\partial p}{\partial t}-\nabla ·\left(K\left(x,p\right)\nabla p\right)=f\left(t,x\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\left(0,T\right]×{\Omega }$

with the initial condition $p\left(0,x\right)={p}^{0}\left(x\right)$ in ${\Omega }$ and Neumann boundary condition $\left(K\left(x,p\right)\nabla p\right)·\nu =g$ on $\left(0,T\right]×{\Gamma }$, where ${\Omega }$ is a rectangular domain in ${ℝ}^{d}$ $\left(1\le d\le 3\right)$, ${\Gamma }$ is its boundary, $\nu$ is the outward unit normal vector on ${\Gamma }$ and $K:{\Omega }×ℝ\to {ℝ}^{d×d}$ is a symmetric, positive definite second-order diagonal tensor.

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 ${L}^{2}\left({\Omega }\right)$ 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 $O\left({H}^{2}+{\Delta }t\right)$ is also given. (Here $d=2$; in general, the order of the error estimation is $O\left({H}^{4-d|2}+{\Delta }t\right)$. 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 $O\left({H}^{4-d|2}+{h}^{2}+{\Delta }t\right)$.

No computational results are given, but they are announced to appear in later papers.

##### MSC:
 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)