Lagrange–Galerkin Methods on Spherical Geodesic Grids

Abstract

Lagrange–Galerkin finite element methods that are high-order accurate, exactly integrable, and highly efficient are presented. This paper derives generalized natural Cartesian coordinates in three dimensions for linear triangles on the surface of the sphere. By using these natural coordinates as the finite element basis functions we can integrate the corresponding integrals exactly thereby achieving a high level of accuracy and efficiency for modeling physical problems on the sphere. The discretization of the sphere is achieved by the use of a spherical geodesic triangular grid. A tree data structure that is inherent to this grid is introduced; this tree data structure exploits the property of the spherical geodesic grid, allowing for rapid searching of departure points which is essential to the Lagrange–Galerkin method. The generalized natural coordinates are also used for determining in which element the departure points lie. A comparison of the Lagrange-Galerkin method with an Euler–Galerkin method demonstrates the impressive level of high order accuracy achieved by the Lagrange–Galerkin method at computational costs comparable or better than the Euler–Galerkin method. In addition, examples using advancing front unstructured grids illustrate the flexibility of the Lagrange–Galerkin method on different grid types. By introducing generalized natural coordinates and the tree data structure for the spherical geodesic grid, the Lagrange–Galerkin method can be used for solving practical problems on the sphere more accurately than current methods, yet requiring less computer time.