Mathematics / advanced
Finite element theory
Derive finite element approximations from weak formulations and prove stability, convergence, and reliable error control.
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Module 01 advanced
Finite element theory
Derive finite element approximations from weak formulations and prove stability, convergence, and reliable error control.
- 01
Weak formulations and Galerkin orthogonality
Derive a finite-dimensional problem from a weak space and identify exactly which residual tests vanish.
30 min - 02
Reference elements and affine maps
Transport basis functions, gradients, and integrals without confusing area scaling with directional distortion.
30 min - 03
Interpolation estimates and Bramble–Hilbert
Combine polynomial reproduction, bounded degrees of freedom, and affine scaling to obtain a genuine approximation estimate.
30 min - 04
Céa’s lemma and quasi-optimality
Separate the space’s approximation capacity from the variational stability constant and the norm in which optimality holds.
30 min - 05
Conforming elements and boundary conditions
Build globally admissible functions and impose essential data through an affine trial set and correct load modification.
30 min - 06
Mixed formulations and inf-sup stability
Control coupled primal and multiplier variables using kernel coercivity and a mesh-uniform coupling condition.
30 min - 07
Saddle-point problems and Stokes elements
Choose velocity and pressure spaces that control incompressibility and eliminate the unavoidable pressure gauge.
30 min - 08
Nonconforming methods and Strang lemmas
Add consistency defects to approximation error when spaces or assembled forms differ from the continuous problem.
30 min - 09
Residual error estimators and reliability
Bound the unknown energy error from element residuals and flux jumps while retaining data oscillation and patch dependence.
30 min - 10
Adaptive refinement and convergence
Connect bulk marking to estimator reduction and distinguish convergence from optimal error per computational cost.
30 min