Course:  Math 272A: Numerical Partial Differential Equations I

Class meet time and location:  TTh 05:00 PM–06:20 PM, APM 5829
Office hours:  TTh 3–4pm or by appointment, APM 5755

Credit Hours:  4
Prerequisite:  Graduate standing or consent of instructor.
Course content:  Finite difference and finite element methods for elliptic partial differential equations. Topics include consistency, stability, and convergence of finite difference schemes; weak formulations and the Lax–Milgram theorem; Sobolev spaces; finite element approximation and a priori error estimates; and a posteriori error estimation and adaptive mesh refinement.

Tentative plan for the Math 272 sequence:
Part A: Finite difference and finite element methods for elliptic PDEs, including adaptive mesh refinement.
Part B: Mixed methods, the Stokes equation, time-dependent PDEs, meshless methods on point clouds, and Lagrangian particle methods.
Part C: Parametric PDEs and neural networks for PDEs.

Recommended texts:

Lecture:   Lectures will be held in person at APM B412. Lectures will also be recorded and made available on Podcast.

Homework:   Homework assignments will be posted on Canvas. There are no individual assignment deadlines, but all homework must be submitted before the final week of the course.

Each homework may contain both analytical and programming problems. You are allowed to use any preferred programming languages. Please read the homework assignments guidelines below.

  • Programming problem requirements:
    1. Describe the method/algorithm used to solve the problem.
    2. Discuss the numerical experiments conducted(specify input/output and the parameters used).
    3. Report the key observations of the numerical experiments. Provide some analysis (performance, results) and offer a conclusion
    4. Use tables/graphics to summarize the data/output from experiments to support the analysis and conclusion.
    5. Attach the computer program written for the project as supplementary materials.
    Note:
    a. Please avoid turning in pages of computer programs output without explanation.
    b. Please provide captions to tables, legends for pictures/plots.
  • Collaboration policy:
    1. It is okay to discuss the problems with others, but you must write your own solutions.
    2. For the programming problems, it is okay to work on the development and debugging with others, but you must do your own runs, make your own plots, etc.
    3. If you worked together with someone on a homework assignment, you must write down who you worked with.
  • Use of AI Tools: AI can be helpful, but only if used responsibly.
    What's allowed:
       1. Review background concepts.
       2. Check explanations or get hints after attempting the problem.
       3. Debug your own code after writing it.
       4. Generate extra practice examples to deepen understanding.
    What’s not allowed:
       1. Submitting AI-generated solutions or code as your own.
       2. Copying and pasting answers directly into your submission.
       3. Using AI to complete homework without understanding the solution.
    If you use an AI tool to help you understand a concept, acknowledge it briefly in your submission (e.g., “I used ChatGPT to clarify the definition of column space.”)
    For more information, visit the UCSD Academic Integrity website.

Grading:  Your grade in the course will be based on class participation (30%), homework (40%), and the final exam (30%).

Academic Integrity:  Academic integrity is highly valued at UCSD and academic dishonesty is considered a serious offense. Students involved in an academic integrity violation will face an administrative sanction which may include suspension or, in very serious cases, expulsion from the university. Your integrity has great value: Cultivate and protect your academic integrity. For more about academic integrity and its value, visit the UCSD Academic Integrity Website.