This collection of notebooks presents a series of general principles and reference implementations for anisotropic Partial Differential Equations (PDEs) in divergence form, using adaptive finite difference schemes on cartesian grids.
where $h$ is a gridscale, $\lambda_i(x)$ is a non-negative weight, and $u(x+h e_i) - u(x)$ is the finite difference of $u$ at positition $x\in \Omega_h$ and in the direction $e_i$. In our implementation, $\Omega_h\subset Z^d$ is a Cartesian grid, and $e_i \in Z^d$ is an offset with integer coordinates, so that $x+h e_i$ is a well defined grid point.
Domain dimension. Unless otherwise specified, the numerical experiments presented in the notebooks involve two dimensional PDEs.
Sobolev spaces. Recall that $H^1(\Omega)$ denotes the set of functions with a square integrable gradient, $H^1_0(\Omega)$ the subspace of functions with null boundary conditions.
Github repository to run and modify the examples on your computer. AdaptiveGridDiscretizations
Main summary, including the other volumes of this work.
3 Automatic differentiation of seismograms 4. Elasticity
#import sys; sys.path.append("..") # Allow imports from parent directory
#from Miscellaneous import TocTools; print(TocTools.displayTOCs('Div'))