A theory for local, a posteriori, pointwise, residual-based estimation of the finite element error

Residual based, a posteriori FEM error estimation is based on the formulation and solution of local, boundary value problems for the error. The error problem is inherently global, but it is split into local problems utilizing a recovery of the local boundary conditions for the error on single elements or patches of elements: “Recovery of the fluxes” or “splitting of the jumps”. Approximation decisions are involved in the splitting of the global problem into local problems, and the actual estimator depends heavily on the type of approximations performed. To provide a foundation for the decision process and to understand the various versions of residual estimators in use today, it is essential to know the underlying mathematical theory behind the error estimation problem, and the restrictions that is put on the problems. In this work, such a theory is presented, and in light of the theory, various error estimators are developed.