Adaptivity of error components for optimality in numerical simulations

• Date: 22. 10. 2026
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K3
• Speaker: Martin Vohralík

In numerical approximation of partial differential equations, several key choices are to be made: 1) physical model, i.e., partial differential equation and its parameters; 2) approximation space given by the mesh size, mesh grading, and approximation polynomial degree; 3) iterative linearization scheme and its precision; 4) iterative algebraic linear solver and its precision. Consequently, different error components arise, and it is known that an optimal algorithm balances all of them. This talk shows some guiding principles of the design of optimal adaptive algorithms and of their numerical analysis. The characterization "optimal" means that we prove that the designed algorithms leads to the best, among all possible choices, decay rate of the error with respect to the cost. Here, the cost is expressed as the sum of the arising degrees of freedom over all iterations (algebraic solver, iterative linearization, approximation space, model), which indeed corresponds to the price one pays to obtain the result of a numerical approximation. Numerical examples illustrating the theory are shown, and an application to the geological sequestration of CO2 is discussed.