APRILE

16 Introduction to mixed problems. The mixed formulation of Laplace problem
16 The Stokes problem and its mixed formulation. Abstract framework for mixed formulations
17 Standard and mixed formulations (symmetric case): associated energy and minimization; saddle point
17 Abstract setting: towards the Banach closed range theorem
20 Banach closed range theorem. Existence and uniqueness of the solution of mixed formulations (simplified case)
20 Existence and uniqueness of the solution of mixed formulations (general case: necessary and sufficient conditions)
23 Verification of the sufficient conditions for Laplace and Stokes problems
23 Inf-sup conditions. A priori estimates for the continous solution. The case of B not surjective
24 Finite element approximation of mixed problems. Existence, uniqueness, and stability of the discrete solution (discrete inf-sup and conditions on the discrete matrix)
24 Consistency and stability implies convergence. Convergence and error estimate for the discrete solution. Commuting diagrams
27 Example: discrete divergence and continuous divergence. Error estimates: discrete kernel included in the continuous kernel
27 Mixed finite elements for the Laplace problem: P1-P0 does not work

MAGGIO

 4 Properties of functional spaces defined on the union of two subdomains (continuity of traces for H1 and Hdiv)
 4 Definition of Raviart-Thomas spaces on simplices and first properties
 7 Raviart-Thomas spaces: interpolation, count of degrees of freedom, unisolvence
 7 Introduction to the Piola transform
11 Piola transform and its properties
11 Approximation properties of Raviart-Thomas spaces
14 Global interpoolation for Raviart-Thomas spaces and approximation properties. Fortin operator
14 Inf-sup condition for Raviart-Thomas spaces. Stability and error estimates for the mixed Laplacian approximated via Raviart-Thomas scheme. Introduction to the approximation of the Stokes problem
15 Remarks on P1-P0 and P1-P1 elements. Spurious pressure modes. The P2-P0 element
15 Fortin operator for the P2-P0 element. The SMALL element (2D and 3D). The MINI element
18 Crouzeix-Raviart element. Q2-P1 element and higher order families
18 Taylor-Hood element (sketch on macroelement condition). Overview on the approximation of eigenvalue problems