A partially mesh-free scheme for representing anisotropic spatial variations along field lines
A common numerical task is to represent functions which are highly spatially anisotropic, and to solve differential equations related to these functions. One way such anisotropy arises is that information transfer along one spatial direction is much faster than in others. In this situation, the derivative of the function is small in the local direction of a vector field $\mathbf{b}$. In order to define a discrete representation, a set of surfaces $M_i$ indexed by an integer $i$ are chosen such that mapping along the field $\mathbf{b}$ induces a one-to-one relation between the points on surface $M_i$ to those on $M_{i+1}$. For simple cases $M_i$ may be surfaces of constant coordinate value. On each surface $M_i$, a function description is constructed using basis functions defined on a regular structured mesh. The definition of each basis function is extended from the surface $M$ along the lines of the field $\mathbf{b}$ by multiplying it by a smooth compact support function whose argument increases with distance along $\mathbf{b}$. Function values are evaluated by summing contributions associated with each surface $M_i$. This does not require any special connectivity of the meshes used in the neighbouring surfaces $M$, which substantially simplifies the meshing problem compared to attempting to find a space filling anisotropic mesh. We explore the numerical properties of the scheme, and show that it can be used to efficiently solve differential equations for certain anisotropic problems.