• The final curriculum is described below. In addition, all exercises with solutions (1-6) are parts of the curriculum.

• Week 34: I haven't followed the book, but what you are supposed to know, and approximately where to find it is:
  The procedure \( (D) \rightarrow (V) \rightarrow (V_h) \leftrightarrow Ax=b\) for a one-dimensional problem (Q.3.2.1)
  The abstract formulation:\( \text{Find $u\in V$ s.t. } a(u,v)=F(v), \forall v\in V\). Properties on \(V\), \(a\) and \(F\). (Q.2.1)
  Lax-Milgram Lemma (Q.3.4.1) In parituclar, understand under which conditions this theorem is valid.
  Cea Lemma (Q. 4.2)
  The minimization problem and Lebesgue and Sobolev spaces in 1D. (Q: 3.2.1)
• Week 35: FEM on 1D-problem: How to set up the extended linear system \(\tilde{A}_h \tilde{u}_h = \tilde{b}_h\).
  The finite element spaces \(x_h^r\) and its lagrangian basis \( \{\varphi_i\} \). Reference element \( \hat{K} \), shape functions on \( \hat{K}\) and the mapping \(\Phi_k: \hat{K} \rightarrow K_k\). How to construct the element matrix and the element load vector. The assembly process. Most of this is covered in this note, see also Q.4.3-4.3.3.
• Week 36: The a priori error estimate for the FEM with \( V_h=X_h^1 \), that is, the interpolation error and the application of Cea's lemma. With proof. (Q:4.3.4-4.3.5). Try to get an idea of how the proof can be extended to cover \( X_h^r \) for \(r>1\).
  Sobolov-spaces (Q.2.4), definitions, inner product and norms. Properties 2.3 (regularity of the spaces), Property 2.4 (Poincare's inequality) and Property 2.5 (equivalence of seminorm and norm on \( H_0^k(\Omega) \). The trace operator. The weak formulation of Poisson's problem (Q:3.3.1, 3.3.3), and use of Lax-Milgram to prove existence and uniqueness of the solution.
• Week 37: The finite element method in \(\mathbb{R}^d\). Triangulation \( \mathcal{T}_h\) (in the beginning of Q:4.5), the abstract definition of a finite element (Q:4.4.1). How to map functions from a reference element \( \hat{K}\) to the physical element \( K\).
• Week 38: Barycentric coordinates (Q:4.4.3), numerical quadrature (Project note, to appear), FEM for the Poisson problem (Q:4.5.1). Triangular finite elements (B&S:3.1-3.2). Conforming FE-spaces (\(X_h \subset V\) ).The following theorem (Braess, 2007):
  Let \(k \geq 1\) and suppose \(\Omega\) is bounded. Then a piecewise infinitely differentiable function \(v:\bar{\Omega}\rightarrow \mathbb{R}\) belongs to \(H^k(\Omega)\) if and only if \(v\in C^{k-1}(\bar{\Omega})\).
• Week 39: A-priori error estimates in the energy norm (Theorem 4.6) and all the bits and pieces of the proof (Q:4.5.3). The \(L^2\) error estimate (Theorem 4.7), with the proof for the Poisson case. Understand the idea of elliptic regularity (Lemma 4.6), and be aware of some counterexamples (BS: 5.5, Example 5.5.2 and 5.5.4).
• Week 40: Parabolic equations: How to set up the weak formulation. Time integration of the space-discretized system (Q: 5-5.1). Stability of the backward Euler method (first half of Q: 5.4). Extend this to the \(\theta\)-method.
  Eigenvalues and eigenfunctions of the bilinear form \(a(\cdot , \cdot)\), and how to approximate it (the second half of Q:5.4). Condition number of the mass matrix \(M_h\) and the stiffness matrix \( A_h\), (See also Q:4.5.2). The following note by Einar Rønquist may also be useful.
• Week 41: A very rough introduction to linear elasticity theory and vibration, see the note. A bit more information on this topic can be found on It's Learning.
  How to construct unstructured grids: Advancing front technique, Delaunay triangulation (what it is, its nice properties), regularization techniques as diagonal swapping and node displacement. (Q:6.4-6.5).
• Week 42: Some idea about how grid generation in 1D (including boundaries of 2D-problems) can be done. And what is a structured grid, and how can we map a regular grid to a domain. Advantages and disadvantages of structured vs. unstructured grids (Q:6.1-6.3). The properties of a triangulation is given in the beginning of (Q:6.2), we have discussed it before, but this may be a good time for a repetition.
• Week 43: Practical error estimation. A priori estimates, where \(|u|_{H^2(K)}^2\) is estimated based on a reconstruction technique (Q.4.6.1). Some idea about the residual based a posteriori estimate (Q:4.6.2), no details required. Understand how error estimates are used to adapt the mesh.
• Week 44: Conjugate gradient (CG) method and its properties. Preconditioning, why it is needed and how it can be done. (Q:7.2.2). THE GMRES method (Q:7.2.3).
• Week 45: Diffusion-transport-reaction problems: Weak formulation, existence and uniqueness results (Q:12.1). Stability analysis of the 1-D problem, Peclet numbers (Q.12.2), artificial diffusion and upwind schemes (Q.12.6).
• Week 46: Exponential fitting (Q.12.6), explained as a Petrov-Galerkin method (Q.12.8.2). Streamline diffusion for the 2D case (Q.12.8.3, strictly speaking not part of the curriculum, but quite useful). Similar analysis and remedies for the diffusion-reaction problem (Q:12.3-12.5, self study).

The curriculum is taken from A. Quarteroni (Q), Numerical Models for Differential Problems, Springer 2014.
We will also use some material from Brenner & Scott (BS): The Mathematical Theory for Finite Element Methods, Springer 2008.

The topics included are:

• Introduction
• The Poisson equation:
  • Weak formulation
  • Finite element method
  • Implementation
  • Error analysis
• Finite element function spaces
• Abstract formalism
• Time dependent problems
• Grid generation
• Error estimation and adaptivity
• Diffusion-transport-reaction problems.

The topics will be described as we proceed.
The schedule should not be taken very literally.