A partially mesh-free scheme for representing anisotropic spatial variations along field lines: Conservation, quadrature, and the delta property

A method for representing highly anisotropic fields is presented, based on a partially meshfree Galerkin formulation. A mapping function is used to provide information about the local direction of the anisotropy, with one of the global coordinates chosen to parameterize the ‘parallel’ position along the mapping in a one-to-one manner. Standard unstructured finite element meshes are used on planes of constant parallel coordinate to represent the necessary small-scale variations perpendicular to the mapping direction, with large spacings then possible between these planes because of the large-scale variations along the mapping. This greatly reduces the number of degrees of freedom required to represent fields in this space and the associated computational cost of simulations involving such fields. No mesh connectivity is defined between planes, and field aligned basis functions are constructed using the mapping function to extend the standard finite element basis into the full domain. Integration of the basis has been addressed with reference to methods developed for fully meshfree methods, and the scheme (as well as other similar element-free Galerkin schemes) is shown to be locally conservative under certain conditions. Robust convergence of several test problems is demonstrated.