Course: The Finite Element Method for Problems in Physics
Modules:
01.01. Introduction. Linear elliptic partial differential equations - I, 01.02. Introduction. Linear elliptic partial differential equations - II, 01.03. Boundary conditions, 01.04. Constitutive relations, 01.05. Strong form of the partial differential equation. Analytic solution, 01.06. Weak form of the partial differential equation - I, 01.07. Weak form of the partial differential equation - II, 01.08. Equivalence between the strong and weak forms, 01.08ct.1. Intro to C++ (running your code, basic structure, number types, vectors), 01.08ct.2. Intro to C++ (conditional statements, “for” loops, scope), 01.08ct.3. Intro to C++ (pointers, iterators)
02.01. The Galerkin, or finite-dimensional weak form, 02.01q. Response to a question, 02.02. Basic Hilbert spaces - I, 02.03. Basic Hilbert spaces - II, 02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation, 02.04q. Response to a question, 02.05. Basis functions - I, 02.06. Basis functions - II, 02.07. The bi-unit domain - I, 02.08. The bi-unit domain - II, 02.09. The finite dimensional weak form as a sum over element subdomains - I, 02.10. The finite dimensional weak form as a sum over element subdomains - II, 02.10ct.1. Intro to C++ (functions), 02.10ct.2. Intro to C++ (C++ classes)
03.01. The matrix-vector weak form - I - I, 03.02. The matrix-vector weak form - I - II, 03.03. The matrix-vector weak form - II - I, 03.04. The matrix-vector weak form - II - II, 03.05. The matrix-vector weak form - III - I, 03.06. The matrix-vector weak form - III - II, 03.06ct.1. Dealii.org, running deal.II on a virtual machine with Oracle VirtualBox, 03.06ct.2. Intro to AWS, using AWS on Windows, 03.06ct.2c. In-Video Correction, 03.06ct.3. Using AWS on Linux and Mac OS, 03.07. The final finite element equations in matrix-vector form - I, 03.08. The final finite element equations in matrix-vector form - II, 03.08q. Response to a question, 03.08ct. Coding assignment 1 (main1.cc, overview of C++ class in FEM1.h)
04.01. The pure Dirichlet problem - I, 04.02. The pure Dirichlet problem - II, 04.02c. In-Video Correction, 04.03. Higher polynomial order basis functions - I, 04.03c0. In-Video Correction, 04.03c1. In-Video Correction, 04.04. Higher polynomial order basis functions - I - II, 04.05. Higher polynomial order basis functions - II - I, 04.06. Higher polynomial order basis functions - III, 04.06ct. Coding assignment 1 (functions: class constructor to “basis_gradient”), 04.07. The matrix-vector equations for quadratic basis functions - I - I, 04.08. The matrix-vector equations for quadratic basis functions - I - II, 04.09. The matrix-vector equations for quadratic basis functions - II - I, 04.10. The matrix-vector equations for quadratic basis functions - II - II, 04.11. Numerical integration -- Gaussian quadrature, 04.11ct.1. Coding assignment 1 (functions: “generate_mesh” to “setup_system”), 04.11ct.2. Coding assignment 1 (functions: “assemble_system”)
05.01. Norms - I, 05.01c. In-Video Correction, 05.01ct.1. Coding assignment 1 (functions: “solve” to “l2norm_of_error”), 05.01ct.2. Visualization tools, 05.02. Norms - II, 05.02. Response to a question, 05.03. Consistency of the finite element method, 05.04. The best approximation property, 05.05. The "Pythagorean Theorem", 05.05q. Response to a question, 05.06. Sobolev estimates and convergence of the finite element method, 05.07. Finite element error estimates
06.01. Functionals. Free energy - I, 06.02. Functionals. Free energy - II, 06.03. Extremization of functionals, 06.04. Derivation of the weak form using a variational principle
07.01. The strong form of steady state heat conduction and mass diffusion - I, 07.02. The strong form of steady state heat conduction and mass diffusion - II, 07.02q. Response to a question, 07.03. The strong form, continued, 07.03c. In-Video Correction, 07.04. The weak form, 07.05. The finite-dimensional weak form - I, 07.06. The finite-dimensional weak form - II, 07.07. Three-dimensional hexahedral finite elements, 07.08. Aside: Insight to the basis functions by considering the two-dimensional case, 07.08c In-Video Correction, 07.09. Field derivatives. The Jacobian - I, 07.10. Field derivatives. The Jacobian - II, 07.11. The integrals in terms of degrees of freedom, 07.12. The integrals in terms of degrees of freedom - continued, 07.13. The matrix-vector weak form - I, 07.14. The matrix-vector weak form II, 07.15.The matrix-vector weak form, continued - I, 07.15c. In-Video Correction, 07.16. The matrix-vector weak form, continued - II, 07.17. The matrix vector weak form, continued further - I, 07.17c. In-Video Correction, 07.18. The matrix-vector weak form, continued further - II, 07.18c. In-Video Correction
08.01. Lagrange basis functions in 1 through 3 dimensions - I, 08.01c. In-Video Correction, 08.02. Lagrange basis functions in 1 through 3 dimensions - II, 08.02ct. Coding assignment 2 (2D problem) - I, 08.03. Quadrature rules in 1 through 3 dimensions, 08.03ct.1. Coding assignment 2 (2D problem) - II, 08.03ct.2. Coding assignment 2 (3D problem), 08.04. Triangular and tetrahedral elements - Linears - I, 08.05. Triangular and tetrahedral elements - Linears - II
09.01. The finite-dimensional weak form and basis functions - I, 09.02. The finite-dimensional weak form and basis functions - II, 09.03. The matrix-vector weak form, 09.03c. In-Video Correction, 09.04. The matrix-vector weak form - II, 09.04c. In-Video Correction
10.01. The strong form of linearized elasticity in three dimensions - I, 10.02. The strong form of linearized elasticity in three dimensions - II, 10.02c. In-Video Correction, 10.03. The strong form, continued, 10.04. The constitutive relations of linearized elasticity, 10.05. The weak form - I, 10.05q. Response to a question, 10.06. The weak form - II, 10.07. The finite-dimensional weak form - Basis functions - I, 10.08. The finite-dimensional weak form - Basis functions - II, 10.09. Element integrals - I, 10.09c. In-Video Correction, 10.10. Element integrals - II, 10.11. The matrix-vector weak form - I, 10.12. The matrix-vector weak form - II, 10.13. Assembly of the global matrix-vector equations - I, 10.14. Assembly of the global matrix-vector equations - II, 10.14c. In Video Correction, 10.14ct.1. Coding assignment 3 - I, 10.14ct.2. Coding assignment 3 - II, 10.15. Dirichlet boundary conditions - I, 10.16. Dirichlet boundary conditions - II
11.01. The strong form, 11.01c In-Video Correction, 11.02. The weak form, and finite-dimensional weak form - I, 11.03. The weak form, and finite-dimensional weak form - II, 11.04. Basis functions, and the matrix-vector weak form - I, 11.04c In-Video Correction, 11.05. Basis functions, and the matrix-vector weak form - II, 11.05. Response to a question, 11.06. Dirichlet boundary conditions; the final matrix-vector equations, 11.07. Time discretization; the Euler family - I, 11.08. Time discretization; the Euler family - II, 11.09. The v-form and d-form, 11.09ct.1. Coding assignment 4 - I, 11.09ct.2. Coding assignment 4 - II, 11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I, 11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II, 11.11c. In-Video Correction, 11.12. Modal decomposition and modal equations - I, 11.13. Modal decomposition and modal equations - II, 11.14. Modal equations and stability of the time-exact single degree of freedom systems - I, 11.15. Modal equations and stability of the time-exact single degree of freedom systems - II, 11.15q. Response to a question, 11.16. Stability of the time-discrete single degree of freedom systems, 11.17. Behavior of higher-order modes; consistency - I, 11.18. Behavior of higher-order modes; consistency - II, 11.19. Convergence - I, 11.20. Convergence - II
12.01. The strong and weak forms, 12.02. The finite-dimensional and matrix-vector weak forms - I, 12.03. The finite-dimensional and matrix-vector weak forms - II, 12.04. The time-discretized equations, 12.05. Stability - I, 12.06. Stability - II, 12.07. Behavior of higher-order modes, 12.08. Convergence, 12.08c. In-Video Correction
Conclusion, and the Road Ahead