ETH_NumericalMethodsForPDE

links - NumPDE

website: https://people.math.ethz.ch/~grsam/NUMPDEFL/
script: https://people.math.ethz.ch/~grsam/NUMPDEFL/NUMPDE.pdf

Info

exam
- NumPDE midterm (@2026-04-17)
- NumPDE endterm (@2026-05-29)
exercises
practice classes
- ETH_NumericalMethodsForPDE_Übungen

Hilfsmittel NumericalMethodsForPartialDifferentialEquations

midterm: probably none
endterm: probably none

Exercises

-> FS2026_tasks

Week 1 (16.02.-20.02.): 4 h 45 min

Problem 0-1: Setting up implementation environment for NumPDE course (60 min)
Problem 0-2: Using the GDB Debugger (80 min)
Problem 1-1: Quadratic Functionals (100 min)
Problem 1-3: L-infinity Norms are Bounded by H1-seminorms in 1D (45 min)

Videos

week (16.02.-20.02.):

1.2.1 Elastic Membranes (17 min)
1.2.2 Electrostatic Fields (6 min)
1.2.3 Quadratic Minimization Problems (21 min)
1.3 Sobolev spaces (22 min)
1.4 Linear Variational Problems (21 min)

week (23.02.-27.02.):

1.5 Equilibrium Models: Boundary Value Problems (22 min)
1.6 Diffusion Models: Stationary Heat Conduction (5 min)
1.7 Boundary Conditions (07.35 min)
1.8 Second-Order Elliptic Variational Problems (13 min)
1.9 Essential and Natural Boundary Conditions (18 min)

week (02.03.-06.03):

2.2 Principles of Galerkin Discretization (25 min)
2.3 Case Study: Linear FEMfor Two-Point Boundary Value Problems (24 min)
2.4 Case Study: Triangular Linear FEMin Two Dimensions I (28 min)
2.4 Case Study: Triangular Linear FEMin Two Dimensions II (32 min)

week (09.03.-13.03):

2.5 Building Blocks of General Finite Element Methods (22 min)
2.6 Lagrangian Finite Element Methods (23 min)
2.7.2 Mesh Information and Mesh Data Structures (21 min)
2.7.4 Assembly Algorithms (26 min)
2.7.5 Local Computations (26 min)
2.7.6 Treatment of Essential Boundary Conditions (21 min)

week (16.03.-20.03):

2.8 Parametric Finite Element Methods I (22 min)
2.8 Parametric Finite Element Methods II (27 min)
3.1 Abstract Galerkin Error Estimates (20 min)
3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM (26 min)

week (23.03.-27.03.):

3.3 A Priori (Asymptotic) Finite Element Error Estimates I (10 min)
3.3 A Priori (Asymptotic) Finite Element Error Estimates II (18 min)
3.3 A Priori (Asymptotic) Finite Element Error Estimates III (16 min)
3.4 Elliptic Regularity Theory (11 min)
3.5 Variational Crimes (15 min)
3.8 Validation and Debugging of Finite Element Codes (13 min)

week (30.03.-03.04.):

9.2.1 Heat Equation (10 min)
9.2.2 Heat Equation: Spatial Variational Formulation (8 min)
9.2.3 Stability of Parabolic Evaolution Problems (10 min)
9.2.4 Spatial Semi-Discretization: Method of Lines (6 min)
9.2.7 Timestepping for Method-of-Lines ODE (30 min)
9.2.8 Fully Discrete Method of Lines: Convergence (18 min)

week (13.04.-17.04.):

10.1.1 Modeling Fluid Flow (7 min)
10.1.2 Heat Convection and Diffusion (5 min)
10.1.3 Incompressible Fluids (10 min)
10.1.4 Time-Dependent (Transient) Heat Flow in a Fluid (4 min)
10.2.1 Singular Perturbation (16 min)
10.2.2 Upwinding (13 min)
10.2.2.1 Upwind Quadrature (17 min)
10.2.2.2 Stremline Diffusion (19 min)
10.3.1 Method of Lines (13 min)

week (20.04.-24.04.):

12.1 Modeling the Flow of a Viscous Fluid (30 min)
12.2.1 The Stokes Equations: Constrained variational formulation (20 min)
12.2.2 The Stokes Equations: Saddle Point Problem Formulation (55 min)
12.2.3 Stokes System of Partial Differential Equations (15 min)

week (27.04.-01.05.):

12.3.1 Pressure Instability (40 min)
12.3.2 Stable Galerkin Discretization of Stokes Saddle Point Problem (30 min)
12.3.3 Convergence of Stable FEM for Stokes (30 min)
12.3.4 The Taylor-Hood Finite Element Method (15 min)
12.3.5 The Non-Conforming Crouzeix-Raviart FEM (40 min)

week (04.05.-08.05.):

13.1.2 Fields and Integral Forms/Currents (28 min)
13.1.3 Differential Forms (56 min)
13.1.4 Exterior Calculus (70 min)
13.1.5 Sobolev Spaces of Differential Forms (18 min)

week (11.05.-15.05.):

13.2.1 Cochain Calculus (67 min)
13.2.2 Whitney Forms (70 min)
13.3.1 Energies and Linear Material Laws (35 min)
13.3.2.1 Electromagnetic Wave Equations (45 min)
13.3.2.3 Method of lines for Electromagnetic Wave Equations, Timestepping for Semi-Discrete Electromagnetic Wave Equations (45 min)

week (18.05.-22.05.):

13.3.3.1 Magnetostatics: Formulations Based on Scalar Potentials (30 min)
13.3.3.2 Magnetostatics: Vector-Potential-Based Variational Formulations (40 min)
13.3.3.3-13.3.3.4 Saddle-Point Variational Problems, Discrete LBB-Conditions for a-Based Magnetostatics (60 min)

week (25.05.-29.05.): no videos

Vorlesung

Video 1.2.2 Eloectrostatic Fields (6min)

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Note: $\hat{V} \subset C^{0} (\overset{―}{Ω})$
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Video 1.2.3 Quadratic Minimization Problems (21 min)

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extension to affine space:

\hat{V} = \underset{"offset function"}{\underset{⏟}{μ_{0}}} + V_{0} \subset V

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Recall from LA:

A = A^{⊤} s.p.d ⟹ (\vec{μ}, \vec{η}) \to {\vec{μ}}^{⊤} A \vec{η} \hat{=} inner product

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Tldr

we are trying to find the most stable/lowest energy state of the system. The idea is to find a general formula for the structurally similar string/membrane/electricity problem, which can be described by a Quadratic functional:

J (v) = \frac{1}{2} a (v, v) - L (v) = \frac{1}{2} x^{⊤} A x - b^{⊤} x (a bilinear, L linear form)

However, we can only solve the problem if the energy formula has a minimum (imagine the energy formula as a paraboloid "bowl"):
Pasted image 20260217110348.png|300
Thus, the