Phelm Documentation

1.0

Introduction

This library is meant to solve partial differential equations. It implements the Finite Element Method. The library is able to solve partial differential equations on two-dimensional smooth manifolds.

To solve a problem you need the following:

triangulate the manifold;
partition the manifold into subdomains;
specify a local coordinate system in each subdomain;
specify a function of surface integral in the local coordinates.

The library already contains triangulation builders for spherical surfaces and flat rectangular domains.

The sample result of mesh builder is shown below:

In that case the sphere is built on 4 parts.

Usage examples

These examples are to give you some tips on Phelm features.

Laplace equation on a flat domain
$\begin{eqnarray*} \Delta u &=& f(x, y) \\ u|_{\partial\Omega}&=&u_0 \end{eqnarray*}$
Laplace equation on a sphere
$\begin{eqnarray*} \Delta \psi &=& f(\varphi, \lambda) \\ \Delta \psi &=& \frac{1}{cos\varphi}\frac{\partial}{\partial\varphi}cos(\varphi)\frac{\partial}{\partial\varphi}\psi+ \frac{1}{cos^2\varphi}\frac{\partial^2}{\partial\lambda^2}\psi\\ \psi|_{\partial\Omega}&=&\psi_0 \\ \end{eqnarray*}$
Double Laplace equations on a flat domain
$\begin{eqnarray*} \Delta u + v &=& f(x, y)\\ u + \Delta v &=& g(x, y)\\ u|_{\partial\Omega}&=&u_0\\ v|_{\partial\Omega}&=&v_0\\ \end{eqnarray*}$
Chafe-Infante equation on a flat domain
$\begin{eqnarray*} \frac{du}{dt} &=& \mu \Delta u - \sigma u + f (u) \\ u(x,y,t)|_{\partial\Omega}&=&a \\ u(x,y,t)|_{t=0} &=& u_0 \\ \end{eqnarray*}$
Chafe-Infante equation on a sphere
The Barotropic vorticity equation on a sphere
$\begin{eqnarray*} \frac{\partial \Delta \varphi}{\partial t} + J(\psi, \Delta \psi) + J(\psi, l + h) + \sigma \Delta \psi - \mu \Delta^2 \psi &=& f(\varphi, \lambda) \\ \psi|_{t=0}=\psi_0 \end{eqnarray*}$
The two-dimensional baroclinic atmosphere equations on a sphere
$\begin{eqnarray*} \frac{\partial \Delta u_1}{\partial t} + J(u_1, \Delta u_1 + l + h) + J(u_2, \Delta u_2) + \frac{\sigma}{2} \Delta (u_1 - u_2) - \mu \Delta^2 u_1 &=& f(\phi, \lambda)\\ \frac{\partial \Delta u_2}{\partial t} + J(u_1, \Delta u_2) + J(u_2, \Delta u_1 + l + h) + \frac{\sigma}{2} \Delta (u_1 + u_2) - \mu \Delta^2 u_2 &-&\\ -\alpha^2 (\frac{\partial u_2}{\partial t} + J(u_1, u_2) - \mu_1 \Delta u_2 + \sigma_1 u_2 + g(\phi, \lambda)) &=& 0,\\ u_1|_{t=0}&=&u_{10}\\ u_2|_{t=0}&=&u_{20}\\ \end{eqnarray*}$