allen\_cahn\_2D module
======================

Solving 2D Allen-Cahn eqations in 2D with homogenous boundary conditions


* Consturct V_n := span { \phi_{n}}, where \phi_{n} is n-th order Polynomial such that \phi_{n} = 0 at the boundaries

* Consturct \phi_{n} = L_{n+2} - L_{n} , where L_{n} is n-th order Legendre Polynomial

* The Quad can be computed as difference of two quad, using linearity as,  

    Consider ( f , \phi_{j} )  =   (  f , L_{j+2}  )   -   (  f , L_{j}  )

* For mass matrix : in the above quad rule plug  : f = L_{k+2}  -  L_{k}

* For 2D Quad, constrct :    \phi_{ij}  =    \phi_i o  \phi_j

    Consider, (f (x, y ), \phi_{ij} )   =   ( f (x) o f(y) , \phi_{ij} )  =  ( f(x) ,  \phi_{i})  o  ( f(y) ,  \phi_{j})


.. automodule:: allen_cahn_2D
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