youwuyou/NPDECODES

401-0674-00L Numerical Methods for Partial Differential Equations FS2023

C++MIT

Exercise Overview

Exam-Relevant

Chapter 1: Second-Order Scalar Elliptic Boundary Value Problems (FS2023)

Theoretical exercises

Number	Topic	Finished
Problem 1-1	Quadratic Functionals	✔️
Problem 1-2	Linear functionals on Sobolev spaces	✔️
Problem 1-3	L∞-Norms Are Bounded by H1	✔️
Problem 1-4	A Poincaré-type inequality
Problem 1-5	A second-order elliptic transmission problem in 1D	✔️
Problem 1-6	Heat conduction with non-local BDC	✔️
Problem 1-7	A second-order boundary value problem for vector fields	✔️
Problem 1-8	A coupled reaction-diffusion problem	✔️

Midterm/Endterm

Number	Topic	Finished
Problem 1-9	Second-order Elliptic BVP from weak formulations	✔️
Problem 1-10	From BVP to Variation Problem	✔️

Chapter 2: Finite Element Methods (FS2023)

Number	Topic	Finished
Problem 2-1	Properties of Galerkin solutions	✔️
Problem 2-2	Transformation of Galerkin Matrices	✔️
Problem 2-3	Pointwise “Exact” Galerkin Solution	04
Problem 2-4	Linear finite elements for two-point boundary value problems	05
Problem 2-5	Triangular linear FEM for 2D reaction-diffusion BVP	06
Problem 2-6	Incidence matrices of a hybrid 2D mesh
Problem 2-7	Computing the length of the boundary in LEHRFEM++
Problem 2-8	Introduction to local assembly in LEHRFEM++	08
Problem 2-9	Handling degrees of freedom (DOFs) in LEHRFEM++	09
Problem 2-10	Projection onto Gradients
Problem 2-11	Hybrid-mesh Galerkin matrices and right-hand-side vectors	07
Problem 2-12	Testing built-in quadrature rules of LEHRFEM++
Problem 2-13	Local computations for parametric Lagrangian finite elements
Problem 2-14	Non-conforming Crouzeix-Raviart FEM	11
Problem 2-15	Regularized Neumann Problem
Problem 2-16	Rigidity of Piecewise Polynomial Continuous Functions	10

Midterm/Endterm

Number	Topic	Finished
Problem 2-17	Local Computations for Convection Bilinear Form
Problem 2-18	Nitsche’s Method for Elliptic BVPs (Winter 2022)	✔️
Problem 2-19	Lagrangian Finite Elements on Criss-Cross Meshes
Problem 2-20	DofHandler and Assembly

Chapter 3: FEM - Convergence and Accuracy (FS2023)

Number	Topic	Finished
Problem 3-1	Computing Averages over the Boundary	13
Problem 3-2	Debugging Finite Element Codes
Problem 3-3	Dirichlet BVP with point-evaluation right-hand-side functional
Problem 3-4	A BVP modelling stationary heat conduction
Problem 3-5	Error estimates for traces	✔️
Problem 3-6	Projection Onto Constants	12
Problem 3-7
Problem 3-8	Output Functionals for a 2nd-Order Elliptic Boundary Value Problem with Impedance Boundary Conditions (Summer 2019)
Problem 3-9	Zienkiewicz-Zhu A-Posteriori Error Estimator (Winter 2020)
Problem 3-10	Parametric Finite Elements
Problem 3-11
Problem 3-12
Problem 3-13	Computation of Stationary Currents (Summer 2020)	14
Problem 3-14	A Local Quasi-Interpolation Operator (Winter 2021)

Midterm/Endterm

Number	Topic	Finished
Problem 3-15	Asymptotic Convergence of FE Discr. and Interp. Errors
Problem 3-16	Non-conforming Crouzeix-Raviart FEM: Theoretical aspects (Summer 2021)
Problem 3-17	Residual-Based A-Posteriori Error Estimator (Summer 2021)
Problem 3-18	Hierarchical Local A-Posteriori Error Estimator (Winter 2022)	✔️
Problem 3-19	Convergence of Finite-Element Solutions

Chapter 5: Non-Linear Elliptic BVPs (FS2023)

Number	Topic	Finished
Problem 5-1	A curve with tension	15
Problem 5-2	The Brachistochrone Problem
Problem 5-3	Minimal Surface Problem for Graphs	16

Midterm/Endterm

Number	Topic	Finished
Problem 5-4	Iterative Methods for Non-Linear Variational Problems

Chapter 9: Second-Order Linear Evolution Problems (FS2023)

Number	Topic	Finished
Problem 9-1	Implicit Two-Stage Radau RK-SSM for Parabolic IBVPs	18
Problem 9-2	Implicit Timestepping for Parabolic IBVP
Problem 9-3	Decaying Method-of-Lines Solution with Implicit-Euler Timestepping	17
Problem 9-4
Problem 9-5	Symplectic Timestepping for Wave Equations
Problem 9-6	A mixed elliptic-hyperbolic linear evolution problem (Summer 2019)	✔️
Problem 9-7	Absorbing Boundary Conditions (ABCs) for the 2D Wave Equation (Winter 2020)
Problem 9-8	Non-linear Schrödinger Equation with Cubic Non-Linearity	19
Problem 9-9
Problem 9-10	Mixed-Hybrid Finite-Element Method for the Wave Equation (Winter 2021)
Problem 9-11	Gauss-Lobatto IIIC Timestepping (Summer 2020)
Problem 9-12	The Sobolev Evolution Initial-Boundary Value Problem (Summer 2021)

Midterm/Endterm

Number	Topic	Finished
Problem 9-13	1D Wave Equation with Perfectly Matched Layers
Problem 9-14	Two-Step Radau Runge-Kutta Single-Step Method for the Heat Equation

Chapter 10: Convection-Diffusion Problems (FS2023)

Number	Topic	Finished
Problem 10-1	Exponentially fitted upwind scheme	21
Problem 10-2	Upwind Quadrature Method	22
Problem 10-3
Problem 10-4	Method of Characteristics and Semi-Lagrangian Discretization
Problem 10-5	Semi-Lagrangian Discretization of Transport Problem	23

Midterm/Endterm

Number	Topic	Finished
Problem 10-6	Streamline Upwind Method for Pure Advection Problem	24

Chapter 11: Numerical Methods for Conservation Laws (FS2023)

Number	Topic	Finished
Problem 11-1	Burgers Equation	25
Problem 11-2	Conservative finite-volume discretization based on Engquist-Osher numerical flux	26
Problem 11-3	Conservative finite-volume discretization based on Godunov numerical flux
Problem 11-4	Lax-Wendroff Scheme	27
Problem 11-5	Discontinous Galerkin Discretization for 1D Conservation Laws	29
Problem 11-6	One-Dimensional Scalar Conservation Law with Empiric Flux (Summer 2019)	✔️
Problem 11-7	Higher-Order Finite Volume Method (Winter 2020)
Problem 11-8	2D Finite Volume Method for Advection	30
Problem 11-9	Non-Linear Conservation Laws with Source Terms (Summer 2020)
Problem 11-10	Flux-Limited Finite-Volume Methods for Scalar Conservation Laws (Winter 2021)

Midterm/Endterm

Number	Topic	Finished
Problem 11-10	Flux-Limited Finite-Volume Methods for Scalar Conservation Laws	28
Problem 11-11	Conservation laws with non-linear density (Winter 2022)	✔️

Exam-Irrelevant

Chapter 4: Beyond FEM - Alternative Discretizations

Number	Topic	Finished
Problem 4-1	Mehrstellenverfahren for Poisson Equation

Chapter 6: Numerical Integration - Single Step Methods

Number	Topic	Finished
Problem 6-1	Linear ODE in spaces of matrices
Problem 6-2	Explicit Runge-Kutta Methods
Problem 6-3
Problem 6-4	System of Second-Order ODEs
Problem 6-5
Problem 6-6
Problem 6-7	Initial Condition for Lotka-Volterra ODE
Problem 6-8
Problem 6-9
Problem 6-10	Symplectic Timestepping for Equations of Motion
Problem 6-11

Chapter 7: Single Step Methods for Stiff Initial Value Problems

Number	Topic	Finished
Problem 7-1	Implicit Runge-Kutta method
Problem 7-2	Damped precession of a magnetic needle
Problem 7-3	Singly Diagonally Implicit Runge-Kutta Method
Problem 7-4	Semi-implicit Runge-Kutta SSM
Problem 7-5
Problem 7-6
Problem 7-7
Problem 7-8