2D Wave Equation (FEM, deal.II)

This project implements a minimal finite element solver for the 2D wave equation

$u_{tt} = c^2 (u_{xx} + u_{yy})$

on the unit square with:

• Homogeneous Dirichlet boundary conditions: $u = 0$ on the boundary.
• Initial displacement: $u(x,y,0) = \sin(\pi x) \sin(\pi y)$.
• Initial velocity: $u_t(x,y,0) = 0$.

The solution is advanced in time using a two–step scheme with assembled mass (M) and stiffness (K) matrices. At the final time, the solver outputs a ParaView‑ready .vtu file and computes the L2 error against the exact analytical solution.

The project also ships with a small smoke‑test harness. Everything is containerized so no local deal.II install is required.

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Run with Docker (recommended)

Only Docker is required. The image build step compiles the code inside the container.

# 1) Clone
git clone https://github.com/<your-user>/<your-repo>.git
cd <your-repo>

# 2) Build the image (this compiles the C++ code inside Docker)
docker build -t fem-wave:1.0 .

# 3) Choose a host folder for outputs (ParaView will open files from here)
mkdir -p outputs

# 4) Run the solver and write all files into ./outputs on your machine
docker run --rm -v "$PWD/outputs:/outputs" -w /outputs fem-wave:1.0 \
  /app/build/wave_solver --h 0.04 --dt 2.5e-4 --c 1.0

# (Optional) Run the tests, logs go to ./outputs
docker run --rm -v "$PWD/outputs:/outputs" -w /outputs fem-wave:1.0 \
  /app/build/run_tests

# (Optional) Or via CTest for verbose failure messages
docker run --rm -v "$PWD/outputs:/outputs" -w /outputs fem-wave:1.0 \
  ctest --test-dir /app/build --output-on-failure

Windows PowerShell: replace $PWD with ${PWD} or use %cd% in CMD.

What gets produced

• final_solution.vtu — final-time displacement/velocity (open in ParaView)
• out*.txt — textual logs written by the test harness

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Command‑line arguments

The solver accepts a few simple flags (all optional with sensible defaults):

• --h <0<h<1> : target mesh size (default 0.025)
• --dt <>0> : time step (default 5e-4)
• --c <>0> : wave speed (default 1.0)

Examples:

# coarser mesh, larger dt
/app/build/wave_solver --h 0.05 --dt 1e-3 --c 1.0

# finer mesh
/app/build/wave_solver --h 0.02 --dt 1.25e-4 --c 1.0

The program validates inputs and exits non‑zero if a value is invalid.

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Tests

We ship a tiny executable run_tests that:

• checks baseline accuracy via a measured L2 error parsed from solver output,
• checks that refinement improves the error,
• verifies invalid inputs are rejected (non‑zero exit),
• asserts a small run finishes under a few seconds (in Release).

Run inside Docker (writes out*.txt to ./outputs on the host):

docker run --rm -v "$PWD/outputs:/outputs" -w /outputs fem-wave:1.0 /app/build/run_tests

Internally, tests call the solver in the container at /app/build/wave_solver. In CMake, we set BIN="${CMAKE_BINARY_DIR}/wave_solver" so the test can locate the solver regardless of cwd.

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License

MIT (see LICENSE).

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Citation / Acknowledgements

• Built with deal.II.
• Sparse direct solves via SparseDirectUMFPACK.