A unified solution method for linear elliptic Cauchy problems

A new method for the solution of Cauchy problems involving linear elliptic operators is introduced. Starting with an initial guess for the missing boundary condition, the algorithm obtains corrections to the assumed value at every iteration. For the updating part, it uses a sampling function to relate the error field to the unknown boundary condition. It uses the data evaluated at points on the boundary. A filtering technique is also introduced which is tailored to the specific problem at hand. The proposed method requires the Green’s function for the linear operator. A number of systems including Laplace, Helmholtz and biharmonic equations are studied. A number of numerical examples are used to show the applicability of the method in the presence of noise.