Monge-Ampére Equation Solver

Introduction

In this repository, we are going to implement two-scale method for Monge-Ampére equation on multi-thread CPU. The equation is:

$$\left\{ \begin{aligned} \det D^2 u = f(x) &\qquad\text{in} &\Omega \subset \mathbb{R}^2, \\\ u = g(x) &\qquad\text{on} &\partial \Omega \end{aligned} \right.$$

where $\Omega$ is a domain in $\mathbb{R}^2$ and $f(x), g(x)$ are given functions. In this repository, we assume that $\Omega$ is a square $[0, 1]^2$, and discretized by a unstructured grid with diagram equals to $h$.

Compile and Run

Install and Compile

This code uses Eigen as the linear algebra library. To compile the code, you need to install Eigen first. To install Eigen, run the following command in the terminal:

git submodule update --init --recursive

To compile the code and run the tests, run the following command in the terminal:

make all

Run Examples

If you want to run the examples without compiling all tests, run the following command in the terminal:

make ./build/example/perron_example_1 && ./build/example/perron_example_1

Following examples is available:

./build/example/perron_example_1
./build/example/perron_example_2
./build/example/perron_example_3
./build/example/newton_example_1
./build/example/newton_example_2
./build/example/newton_example_3

Super parameters can be changed in the corresponding source .cpp file.

Code Structure

.
├── example
│   ├── newton_example_1.cpp    // Example 1 for Newton method
│   ├── newton_example_2.cpp    // Example 2 for Newton method
│   ├── newton_example_3.cpp    // Example 3 for Newton method
│   ├── perron_example_1.cpp    // Example 1 for Perron iteration
│   ├── perron_example_2.cpp    // Example 2 for Perron iteration
│   └── perron_example_3.cpp    // Example 3 for Perron iteration
├── include
│   └── ma.hpp                  // Header file
├── Makefile
├── mesh                        // Mesh files (Typo: 2e-k means h = 2^{-k})
│   ├── 2e-2.obj
│   ├── 2e-3.obj
│   ├── 2e-4.obj
│   ├── 2e-5.obj
│   ├── 2e-6.obj
│   ├── 2e-7.obj
│   └── 2e-8.obj
├── README.md
├── source
│   ├── mesh.cpp                // Mesh struct
│   ├── mesh_function.cpp       // MeshFunction struct
│   ├── point.cpp               // Point struct
│   ├── solver.cpp              // Solver namespace (include CG and GMRES)
│   └── two_scale.cpp           // TwoScale namespace (include Newton and Perron)
└── test
    ├── mesh.cpp                // Test for Mesh struct
    ├── mesh_function.cpp       // Test for MeshFunction struct
    ├── solver.cpp              // Test for Solver namespace
    └── test_main.cpp           // Main function for all tests

freeformgrit / monge-ampere