Syllabus
Basic properties of an a posteriori estimate: guaranteed upper bound, local efficiency, asymptotic exactness, robustness with respect to parameters, low evaluation cost, distinction of error components.
Mathematical framework: continuity of the potential and continuity of the normal trace of the flux: the spaces H1 and H(div), primal and dual variational formulations, Green theorem, Prager and Synge theorem, Poincaré-Friedrichs-Wirtinger inequalities, residual of a partial differential equation, energy norm and dual norms.
Construction and evaluation of the estimators: potential reconstruction, flux reconstruction, equilibration using the mixed finite element method, equivalence with the error.
Theory for model problems: Laplace equation, the advection-diffusion-reaction equation, the Stokes equation, the unsteady heat equation, the nonlinear Laplace equation.
Application to classical numerical methods: conforming finite element method, nonconforming finite element method, mixed finite element method, discontinuous Galerkin method, finite volume method.
Use of the estimates: adaptation of spatial meshes, adaptation of the time step, stopping criteria for linear solvers, stopping criteria for nonlinear solvers.
Annotation
The course treats estimates of the error in approximate numerical solution of partial differential equations. The impact is on guaranteed and fully computable estimates.
A unified framework for classical numerical methods (FEM, DGFEM,...) is introduced. The theory is developed for a large variety of problems.
The emphasis is on the use of the estimates for efficient numerical calculation (adaptive mesh refinement, adaptive choice of the time step, stopping criteria for linear and nonlinear solvers).