Extending Data to Improve Stability and Error Estimates Using Asymmetric Kansa-like Methods to Solve PDEs

In this paper, a theoretical framework is presented for the use of a Kansa-like method to numerically solve elliptic partial differential equations on spheres and other manifolds. The theory addresses both the stability of the method and provides error estimates for two different approximation methods. A Kansa-like matrix is obtained by replacing the test point set $X$, used in the traditional Kansa method, by a larger set $Y$, which is a norming set for the underlying trial space. This gives rise to a rectangular matrix. In addition, if a basis of Lagrange (or local Lagrange) functions is used for the trial space, then it is shown that the stability of the matrix is comparable to the stability of the elliptic operator acting on the trial space. Finally, two different types of error estimates are given. Discrete least squares estimates of very high accuracy are obtained for solutions that are sufficiently smooth. The second method, giving similar error estimates, uses a rank revealing factorization to create a ``thinning algorithm'' that reduces $\#Y$ to $\#X$. In practice, this algorithm doesn't need $Y$ to be a norming set.