---
title: "<Project title>"
subtitle: "AMPS II — Final project report · Topic <A–J or own>"
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

<!-- The application and why it matters for your research (a sketch helps). Geometry, governing equations, boundary/initial conditions,
     parameters WITH UNITS, dimensionless groups, the quantity of interest (QoI). -->

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

| Dimensionless group | Definition | Base case | Range studied |
|---|---|---|---|
| | | | |

# Model and weak form

<!-- Spaces; essential vs natural conditions; constitutive model and internal variables; residual F(u; v); Jacobian (derived, or ufl.derivative
     + how you checked it). The reductions (2D, axisymmetric, plane strain, periodic cell, one-way coupling, steady state) and why each is valid. -->

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

# Numerical choices

<!-- Elements and why they do not lock / satisfy inf-sup; stabilisation (Peclet argument); time stepping and its stability; Newton,
     load or parameter continuation; linear solver and preconditioner; Colab cost. Where can these choices fail for your parameters? -->

# Implementation

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

<!-- Solver options and the reason for each; what the model module contains; what is saved and where it is plotted. -->

# Verification

## Manufactured or exact solution

<!-- u_exact, the conditions it satisfies, PREDICTED rates (with the reason). Errors against the exact EXPRESSION, raised quadrature (Week 15). -->

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

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

<!-- ![Observed convergence with predicted slopes.](figs/rates.png){#fig-rates width=70%} -->

## The topic's physical check

<!-- The check listed on your topic page (or, own topic: a benchmark or analytical limit with a citation [@key]); your value, the deviation in %, on two or more meshes. -->

## Balance or conservation

<!-- Mass / energy / force / flux balance, with the residual relative to the largest term. -->

## Parallel run

<!-- QoI with -np 1 and -np 2 (relative difference). -->

# Results

<!-- The design output and the parameter study. For every reported value: its mesh (and time-step) uncertainty —
     observed order, Richardson value, GCI [@roache1994gci; @celik2008], or three meshes. Physical interpretation. -->

| Case | QoI (finest mesh) | observed order | Richardson / GCI |
|---|---|---|---|
| | | | |

# 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

<!-- What the model and the numerics cannot do; what you would do next. -->

# 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, the topic's physical check, a balance, parallel = serial)?

> …

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

> …