Finite elements on the sphere

Abstract The origin of this paper is the need for methods of solving the so-called altimetry-gravimetry problem of physical geodesy (see Svensson [7]) numerically. In the spherical approximation this problem leads to a pseudodifferential equation on a sphere, involving an invariant (with respect to the Riemannian geometry of the sphere) pseudodifferential operator of order one on the sphere. In trying to apply the Galerkin method with such an operator, it is natural to use trial functions χ P which are of axial symmetry around an axis through the point P . The matrix elements to be constructed in the Galerkin method are then of the form ( Aχ P , χ Q ) and depend, because of the invariance of A and the symmetry of the trial functions, only upon the distance from P to Q . To be able to treat the mixed problem, “small” support for the trial functions is needed. Trial functions χ P satisfying the requirements above are suggested. Approximation theorems are proved. To get approximations compareable to those of the plane finite element approach one has to let the grid size be a power h β , β > 1, of the element diameter h . Efforts of estimating β are made. An analysis of a plane approximation indicates, however, that those estimates are suboptimal. Computation of the matrix elements is discussed.