Finite Element Method in FEniCS: 1D Transient Heat Diffusion in detail

FEM problems can be easily solved in Python by providing the weak form of the PDE as well as the Boundary Condition and Initial Condition to Fenics. In this video, we will go over all steps in detail, incl. Function Spaces and deriving weak forms. Here are the notes: github.com/Ceyron/machine-learning-and-simulation/… This video is an in-detail extension of a previous video, in which we just did a quick and simple demo. Feel free to check this one out first:    • Fenics Tutorial: 1D Heat Conduction   You can find the source-code as written in the video (with slightly more explaining comments): github.com/Ceyron/machine-learning-and-simulation/… Fenics is a software that allows to easily solve Partial Differential Equations in Python. PDEs arise in many disciplines, e.g., in Continuum Mechanics for fluid and solid simulations. ------- 📝 : Check out the GitHub Repository of the channel, where I upload all the handwritten notes and source-code files (contributions are very welcome): github.com/Ceyron/machine-learning-and-simulation 📢 : Follow me on LinkedIn or Twitter for updates on the channel and other cool Machine Learning & Simulation stuff: www.linkedin.com/in/felix-koehler and twitter.com/felix_m_koehler 💸 : If you want to support my work on the channel, you can become a Patreon here: www.patreon.com/MLsim ------- Timestamps: 00:00 Intro 00:48 Initial-Boundary Value Problem 03:48 Initial Condition & Expected Behavior 05:12 Discretization into Finite Elements 06:14 Ansatz/Shape Function 08:16 Discrete PDE solution 11:41 Function Spaces (Lagrange Polynomials) 12:38 Code: Overview 12:51 Code: Mesh Discretization 13:34 Code: Function Space 14:59 Code: Translate IC & BC 19:40 Code Recap 20:02 Why we need the weak form? 21:06 (1) Multiply with test function 21:41 (2) Integrate over domain 22:10 (3) Integration by parts 23:13 What is the test function? 25:21 Vanishing Boundary Evaluation 25:47 Discussing the weak form 26:16 Weak form in residuum form 26:50 Discretization in time 30:27 Fenics wants multi-dim weak form 30:51 Weak form in high dim case 34:08 Multi dimensional integration by parts (divergence theorem) 40:30 Comparison with 1D case 41:25 Summary of high-dim weak form 41:59 Temporal Discretization in high-dim case 42:35 Final Weak Form for Fenics 43:36 Code: Defining Test & Trial Functions 44:38 Code: Weak Form Residuum 47:05 Code: Separate into lhs & rhs 47:49 Code: Time Loop & Simulation 50:20 Code: Adjusting Plot Visuals 51:08 Code: Running & Discussion 51:57 Ou

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