A Posteriori Error Bounds for Linear Elliptic Equations of Potential Type
John E. Lavery · IMA Journal of Applied Mathematics · 1979
Second-order Linear elliptic partial differential equations of potential type with Dirichlet (type 1) or Neumann (type II) boundary conditions on a simply-connected two-dimensional domain are considered. Conjugate problems, that is, a pair of one type 1 and one type II problem, are introduced along with an auxiliary eppiptic system of two equations in such a way that the energies of the given problem, its conjugate problem, and the auxiliary system add to a known constant. There result two-sided bounds for the energy of the given problem and, as a consequence, a posteriori error bounds for the norm of the difference of an approximate solution and the exact solution of the problem. A method by which the amount of computation required to obtain the a posteriori error bounds can be almost halved in many cases of practical interest is given. A posteriori error bounds for approximate solutions of the auxiliary system are also given.