Multigrid for high-order finite elements: line search, p-robustness, a posteriori estimates, and adaptivity

• Date: 28. 4. 2022
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K3
• Speaker: Martin Vohralík

We develop and analyze an iterative solver for large sparse systems of algebraic equations arising from finite element discretizations of high polynomial degree p. It is of multigrid type and has the following characterizations: (1) It is genuinely steered by an a posteriori estimator that certifies the algebraic error in the energy norm. (2) It contracts the algebraic error independently of the polynomial degree p. (3) It is basis independent. (4) The error descent is optimized by a line search on each mesh level. (5) It features an explicit Pythagorean formula for the decrease of the algebraic error along the iterations in terms of level-wise and patch-wise computable error reductions. (6) It is naturally non symmetric: first the roughest modes are captured by the coarse solve, and then smoothing, by additive Schwarz (block-Jacobi), is performed on each mesh level. (7) It is naturally minimalist: only one post-smoothing step is sufficient. (8) It is parameter-free (no damping or number of smoothing steps or other parameters need to be defined). (9) Adaptive number of smoothing steps and adaptive choice of patches to perform smoothing can be straightforwardly set up.