1. Introduction, Linear Elliptic Partial Differential Equations (Part 1)
2. Introduction, Linear Elliptic Partial Differential Equations (Part 2)
3. Boundary Conditions
4. Constitutive relations
5. Strong Form of the Partial Differential Equation, Analytic Solution
6. Weak Form of the Partial Differential Equation (Part 1)
7. Weak Form of the Partial Differential Equation (Part 2)
8. Equivalence Between the Strong and Weak Forms (Part 1)
9. The Galerkin, or finite dimensional weak form
10. Response to a question
11. Basic Hilbert Spaces (Part 1)
12. Basic Hilbert Spaces (Part 2)
13. FEM for the One Dimensional, Linear Elliptic PDE
14. Response to a question
15. Basis Functions (Part 1)
16. Basis Functions (Part 2)
17. The Bi-Unit Domain (Part 1)
18. The Bi-Unit Domain (Part 2)
19. Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 1)
20. Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 2)
21. The Matrix-Vector Weak Form - I (Part 1)
22. The Matrix-Vector Weak Form - I (Part 2)
23. The Matrix-Vector Weak Form - II (Part 1)
24. The Matrix-Vector Weak Form - II (Part 2)
25. The Matrix-Vector Weak Form - III (Part 1)
26. The Matrix-Vector Weak Form - III (Part 2)
27. The Final Finite Element Equations in Matrix-Vector form (Part 1)
28. The Final Finite Element Equations in Matrix-Vector form (Part 2)
29. Response to a question
30. The Pure Dirichlet Problem (Part 1)
31. The Pure Dirichlet Problem (Part 2)
32. Correction to boardwork
33. Higher Polynomial Order Basis Functions - I
34. Correction to boardwork
35. Higher Polynomial Order Basis Functions - 1 (Part 2)
36. Higher Polynomial Order Basis Functions - II (Part 1)
37. Higher Polynomial Order Basis Functions - III
38. The Matrix Vector Equations for Quadratic Basis Functions - I (Part 1)
39. The Matrix Vector Equations for Quadratic Basis Functions - I (Part 2)
40. The Matrix Vector Equations for Quadratic Basis Functions - II (Part 1)
41. The Matrix Vector Equations for Quadratic Basis Functions - II (Part 2)
42. Numerical Integration -- Gaussian Quadrature
43. Norms (Part 1)
44. Correction to boardwork
45. Norms (Part 2)
46. Response to a question
47. Consistency of the Finite Element Method
48. The Best Approximation Property
49. The "Pythagorean Theorem"
50. Response to a question
51. Sobolev Estimates and Convergence of the Finite Element Method
52. Finite Element Error Estimates
53. Functionals, Free Energy (Part 1)
54. Functionals, Free Energy (Part 2)
55. Extremization of Functionals
56. Derivation of the Weak Form Using a Variational Principle
57. The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 1)
58. The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 2)
59. Response to a question
60. The Strong Form, continued
61. Correction to boardwork
62. The Weak Form
63. The Finite Dimensional Weak Form (Part 1)
64. The Finite Dimensional Weak Form (Part 2)
65. Three-Dimensional Hexahedral Finite Elements
66. Aside: Insight to the Basis Functions by Considering the Two-Dimensional Case
67. Field Derivatives: The Jacobian (Part 1)
68. Field Derivatives: The Jacobian (Part 2)
69. The Integrals in Terms of Degrees of Freedom
70. The Integrals in Terms of Degrees of Freedom - Continued
71. The Matrix-Vector Weak Form (Part 1)
72. The Matrix-Vector Weak Form (Part 2)
73. The Matrix-Vector Weak Form, continued (Part 1)
74. Correction to boardwork
75. The Matrix Vector Weak Form, continued (Part 2)
76. The Matrix-Vector Weak Form, continued further (Part 1)
77. Correction to boardwork
78. The Matrix-Vector Weak Form, continued further (Part 2)
79. Correction to boardwork
80. Lagrange Basis Functions in 1 Through 3 Dimensions (Part 1)
81. Lagrange Basis Functions in 1 through 3 dimensions (Part 2)
82. Quadrature Rules in 1 Through 3 Dimensions
83. Triangular and Tetrahedral Elements-Linears (Part 1)
84. Triangular and Tetrahedral Elements Linears (Part 2)
85. The Finite Dimensional Weak Form and Basis Functions (Part 1)
86. The Finite Dimensional Weak Form and Basis Functions (Part 2)
87. The Matrix Vector Weak Form
88. The Matrix Vector Weak Form (Part 2)
89. Correction to boardwork
90. The Strong Form of Linearized Elasticity in Three Dimensions (Part 1)
91. The Strong Form of Linearized Elasticity in Three Dimensions (Part 2)
92. The Strong Form, continued
93. The Constitutive Relations of Linearized Elasticity
94. The Weak Form (Part 1)
95. Response to a Question
96. The Weak Form (Part 2)
97. The Finite-Dimensional Weak Form-Basis Functions (Part 1)
98. The Finite-Dimensional Weak Form-- Basis functions (Part 2)
99. Element Integrals (Part 1)
100. Correction to boardwork
101. Element Integrals (Part 2)
102. The Matrix-Vector Weak Form (Part 1)
103. The Matrix Vector-Weak Form (Part 2)
104. Assembly of the Global Matrix-Vector Equations (Part 1)
105. Assembly of the Global Matrix-Vector Equations II
106. Correction to boardwork
107. Dirichlet Boundary Conditions (Part 1)
108. Dirichlet Boundary Conditions (Part 2)
109. The Strong Form
110. Correction to boardwork
111. The Weak Form, and Finite Dimensional Weak Form (Part 1)
112. The Weak Form, and Finite Dimensional Weak Form (Part 2)
113. Basis Functions, and the Matrix-Vector Weak Form (Part 1)
114. Correction to Boardwork
115. Basis Functions, and the Matrix-Vector Weak Form (Part 2)
116. Response to a question
117. Dirichlet Boundary Conditions; The Final Matrix Vector Equations
118. Time Discretization; The Euler Family (Part 1)
119. Time Discretization; The Euler Family (Part 2)
120. The V-Form and D-Form
121. Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 1)
122. Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 2)
123. Modal Decomposition and Modal Equations (Part 1)
124. Modal Decomposition and Modal Equations (Part 2)
125. Modal Equations and Stability of the Time Exact Single Degree of Freedom Systems (Part 1)
126. Modal Equations and Stability of the Time-Exact Single Degree of Freedom Systems (Part 2)
127. Stability of the Time-Discrete Single Degree of Freedom Systems
128. Behavior of Higher-Order Modes; Consistency (Part 1)
129. Behavior of Higher-Order Modes; consistency (Part 2)
130. Convergence (Part 1)
131. Convergence (Part 2)
132. The Strong and Weak Forms
133. The Finite-Dimensional and Matrix-Vector Weak Forms (Part 1)
134. The Finite-Dimensional and Matri-Vector Weak Forms (Part 2)
135. The Time-Discretized Equations
136. Stability (Part 1)
137. Stability (Part 2)
138. Behavior of High-Order Modes
139. Convergence
140. Correction to boardwork

Here they are then, about 50 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed.

The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based.

It is hoped that these lectures on Finite Element Methods will complement the series on Continuum Physics to provide a point of departure from which the seasoned researcher or advanced graduate student can embark on work in (continuum) computational physics.