---
title: "<Project title>"
subtitle: "AMPS I — Final project report"
author: "<Name> (<student ID>)"
date: "<submission date>"
format:
  html:
    html-math-method: katex
    embed-resources: true
    toc: true
    number-sections: true
    fig-cap-location: bottom
bibliography: references.bib
---

<!-- Keep every section heading. At most 10 printed pages for Sections 1-8. Every number must come from project.ipynb:
     add a source tag after each table or figure caption, e.g. "(project.ipynb §5.2)". Delete these comments before submitting. -->

# Problem

<!-- Physical setting (a sketch helps), governing equations, boundary/initial conditions, parameters WITH UNITS,
     dimensionless numbers, the quantity of interest (QoI). Why it matters for your research. -->

| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| | | | |

# Weak form and discretisation

<!-- Spaces; essential vs natural conditions; residual F(u; v); Jacobian (derived, or ufl.derivative + how you checked it, e.g. a Taylor test).
     Elements, stabilisation, time stepping. -->

$$
F(u;v) = \dots = 0 \quad \forall v\in V_0 .
$$

# Analysis

<!-- Which theorem (Lax-Milgram, Brezzi/inf-sup, monotone operators, CFL/Peclet argument) gives existence, uniqueness or stability of the
     (linearised) problem? Constants. Where can it fail for your parameter range? -->

# Implementation

| Function / cell | Purpose |
|---|---|
| | |

<!-- Solver choices (SNES options, KSP, preconditioner) and the reason for each. -->

# Verification

## Manufactured solution

<!-- u_exact, the boundary conditions it satisfies, PREDICTED rates (with the reason). -->

| $h$ | $\|u-u_h\|_{L^2}$ | rate | $|u-u_h|_{H^1}$ | rate |
|---|---|---|---|---|
| 1/8 | | – | | – |
| 1/16 | | | | |
| 1/32 | | | | |
| 1/64 | | | | |

: MMS convergence; predicted rates $L^2$: …, $H^1$: …. (project.ipynb §5.1) {#tbl-mms}

<!-- ![Observed convergence.](figs/mms_rates.png){#fig-mms width=70%} -->

## Benchmark or analytical check

<!-- Reference value with a citation [@key], your value, deviation in %, on two or more meshes. -->

## Parallel run

<!-- QoI with -np 1 and -np 2. -->

# Results

<!-- QoI, parameter study, mesh (and time-step) independence of the reported values. -->

# LLM workflow

<!-- At least three concrete cases: what was generated, what was wrong, which test exposed it, how you fixed and re-verified it. -->

| # | Component | What the LLM produced | Error | Found by | Fix and re-check |
|---|---|---|---|---|---|
| 1 | | | | | |
| 2 | | | | | |
| 3 | | | | | |

# Limitations and outlook

<!-- Optional Section 9 "Inverse problem" (bonus): Taylor test, finite-difference check, recovery of the true parameter. -->

# References {.unnumbered}

::: {#refs}
:::

## AI-usage log {.unnumbered}

*Required in every assignment and project. Using an LLM is encouraged and never penalized — an empty or vague log is.*

| # | Tool / model | What I asked (key prompt, shortened) | What it got wrong or missed | How I detected it | How I fixed it |
|---|---|---|---|---|---|
| 1 | | | | | |
| 2 | | | | | |
| 3 | | | | | |

**Code provenance** — which cells/functions were mostly generated, which did you write or substantially change?

> …

**Verification** — how do you know the final result is correct (MMS rates, benchmark value, conservation check, parallel = serial)?

> …

**Reflection (2–3 sentences)** — what would you do differently when prompting next time?

> …