Lecture Calendar

04-04-2024  Review of basic results from mathematical analysis (Hilbert spaces, weak derivative, Sobolev spaces, approximation by smooth functions for \(H^1\) functions).

08-04-2024  Trace estimates within a triangle \(K\subset\mathbb{R}^2\) and on \(\Omega\subset\mathbb{R}^2\), where \(\Omega\) a polygonal Lipschitz domain. Riesz Represetantion Theorem (without proof). Lax-Milgram Lemma (without proof).

11-04-2024  Exercises 1-5.

15-04-2024  Proof of Lax-Milgram Lemma. Definition of classical and weak solution of Poisson equation with homogeneous boundary conditions. Proof of the existence and uniqueness of weak solution \(u\in H^1_0(\Omega)\) using Lax-Migram Lemma.

18-04-2024  Dirichlet Principle. Weak form of Poisson equation with non-homogeneous Dirichlet and Neumann conditions. Approximation problem, existence and uniqueness. Error in energy norm.

22-04-2024  Error in energy norm, Cea's Lemma and Aubin-Nitsche Lemma (Proofs). Error in \(H^1\) and \(L_2\) for the Poisson equation with zero boundary conditions where the finite dimensional space \(V_h\) satisfies an approximation property.

25-04-2024  Exercises 6-9.

29-04-2024  Finite element spaces. Construction of finite element spaces. Triangular elements (linear, quadratic Lagrange and cubic Hermite element).

02-05-2024  Rectangular elements (Bilinear, Biquadratic Lagrange Finite Element). Continuous piecewise linear interpolation.

06-05-2024  Piecewise linear, continuous functions on a triangulation. Shape regularity. A local estimate for the interpolant.

13-05-2024  Scaling estimates between functions on reference and generic domain. Proof a local estimate for the interpolant using Bramble-Hilbert lemma in \(L^2\) norm and \(H^1\) seminorm.

14-05-2024  Bramble-Hilbert lemma (Proof). Inverse inequality on quasi-uniform triangulations. Non-conforming finite elements (Motivation).

16-05-2024  Petrov-Galerkin method. Generalized Lax-Milgram Lemma.

23-05-2024  Exercises 10,15,16.

27-05-2024  Error estimates for Petrov-Galerikin method. Introduction to Generalized Galerkin method and its existence and uniqueness.

30-05-2024  First and second Strang Lemma. Application of the Strang lemma (Effects of numerical integration).

03-06-2024  Application of the Strang lemma (Effects of numerical integration). Crouzeix Raviart Element.

06-06-2024  Crouzeix Raviart Element. Error estimation

17-06-2024  Introduction to time dependent problems(Heat equation). Bochner spaces and definition of the weak solution. Construction of the solution via Galerkin method.

18-06-2024  Energy estimates. Existence and uniqueness of the weak solution. For higher regurarity see Chapter 7.1, Reference 8.

20-06-2024  Galerkin method for Heat Equation. Elliptic(Ritz) Projection, \(R_h\,:\,H^1 \to V_h^r,\,r\geq 1,\) integer. Definition and error estimates in \(L^2\) norm and \(H^1\) seminorm.

24-06-2024  Error estimates for the semi-discrete scheme in \(L^2\) norm and \(H^1\) seminorm.

27-06-2024  Fully-discrete scheme with implicit Euler. Matrix formulation and error estimates in \(L^2\) norm.

01-07-2024  Fully-discrete scheme with Crank-Nicolson and explicit Euler. Matrix formulation and error estimates in \(L^2\) norm.

Literature

• [1] Susanne C. Brenner; L. Ridgeway Scott, The mathematical theory of finite element methods, Third Edition, 2008.

• [2] D. Gallistl, Theorie und Numerik partieller Differentialgleichungen I (Computational PDEs I), Lecture notes.

• [3] Philippe G.Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co, 1978

• [4] Alexandre Ern, Jean Luc Guermond, Theory and Practice of Finite Elements, Springer, 2010

• [5] Kendall E. Atkinson, Weimin Han, Theoretical numerical analysis: a functional analysis framework, Springer, 2007

• [6] Mark S. Gockenbach, Understanding and implementing the finite element method, SIAM, Soc. for Industrial and Applied Mathematics, 2006

• [7] Mats G. Larson , Fredrik Bengzon, The Finite Element Method: Theory, Implementation, and Applications , Springer Berlin, Heidelberg, 2015

• [8] Lawrence C. Evans, Partial differential equations, Providence, Rhode Island : American Mathematical Society, 2015

• [9] Vidar Thomée, Galerkin Finite Element Methods for Parabolic Problems, Berlin, Heidelberg : Springer Berlin Heidelberg, 1984

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