they offer increased accuracy and efficiency by virtue of their independence on the CFL condition. Detailed results
Lagrange-Galerkin finite element methods that are high-order accurate, exactly integrable, and highly efficient are presented. This and analyses are given in one-dimension in [13, 14] and in paper derives generalized natural Cartesian coordinates in three two dimensions on the plane in [2,3,17]. Very little work dimensions for linear triangles on the surface of the sphere. By has been done on the weak Lagrange-Galerkin method using these natural coordinates as the finite element basis functions on spherical domains; a thorough review of the literature we can integrate the corresponding integrals exactly thereby achievreveals no published work in this subject. This paper assists ing a high level of accuracy and efficiency for modeling physical problems on the sphere. The discretization of the sphere is achieved in filling this gap in the literature. Spherical geodesic grids by the use of a spherical geodesic triangular grid. A tree data struchave been around for quite some time [20]. However, these ture that is inherent to this grid is introduced; this tree data structure methods have recently been rediscovered [9], as more and exploits the property of the spherical geodesic grid, allowing for more researchers have begun to move away from spectral rapid searching of departure points which is essential to the Lagrange-Galerkin method. The generalized natural coordinates methods toward finite volumes, finite elements, and finite are also used for determining in which element the departure points difference methods for spherical domains.
lie. A comparison of the Lagrange-Galerkin method with an Euler-On spherical geometries, spherical coordinates appear Galerkin method demonstrates the impressive level of high order to be the obvious coordinate system for formulating the accuracy achieved by the Lagrange-Galerkin method at computaproblem; however, there are numerical singularities associtional costs comparable or better than the Euler-Galerkin method.
In addition, examples using advancing front unstructured grids illus-ated with the poles. This issue can be circumvented by trate the flexibility of the Lagrange-Galerkin method on different either rotating the spherical coordinate system as in [11] grid types. By introducing generalized natural coordinates and the or by mapping to Cartesian coordinates whenever we are tree data structure for the spherical geodesic grid, the Lagrangein the vicinity of the poles as is done in [19]. This paper Galerkin method can be used for solving practical problems on the takes a different approach, namely, remaining in Cartesian sphere more accurately than current methods, yet requiring less computer time. ᮊ 1997 Academic Press space throughout [15]. This strategy offers some clear advantages.
First, the spherical geodesic grids are constructed in