Solving the 3D Cauchy problems of nonlinear elliptic equations by the superposition of a family of 3D homogenization functions

摘要

We solve the 3D Cauchy problem of a nonlinear elliptic equation in a cuboid, using the derived family of 3D homogenization functions of different orders. When the solution is expressed by the weight superposition of a family of 3D homogenization functions, the unknown boundary data and the solution can be recovered quickly by solving a small scale linear system. It deserves to note that the superposition of homogenization functions method (SHFM) does not need to solve nonlinear equations and regularization, and is quite accurate to find the solution in the whole domain with the errors smaller than the level of noise being imposed on the overspecified Neumann data. Another advantage of the SHFM is that it can solve the Cauchy problem in a large size of the cuboid.