""" Poisson problem. """


# ===================================================== #
#                                                       #
#  This script implements the two dimensional           #
#  Poisson problem. The Poisson class is defined        #
#  inheriting from SpatialProblem. We  denote:          #
#           u --> field variable                        #
#           x,y --> spatial variables                   #
#                                                       #
# ===================================================== #


import torch
from pina.domain import CartesianDomain
from pina import Condition
from pina.problem import SpatialProblem
from pina.operators import laplacian
from pina.equation import FixedValue, Equation


class Poisson(SpatialProblem):
    output_variables = ['u']
    spatial_domain = CartesianDomain({'x': [0, 1], 'y': [0, 1]})

    def laplace_equation(input_, output_):
        force_term = (torch.sin(input_.extract(['x'])*torch.pi) *
                        torch.sin(input_.extract(['y'])*torch.pi))
        nabla_u = laplacian(output_.extract(['u']), input_)
        return nabla_u - force_term

    conditions = {
        'gamma1': Condition(
            location=CartesianDomain({'x': [0, 1], 'y':  1}),
            equation=FixedValue(0.0)),
        'gamma2': Condition(
            location=CartesianDomain({'x': [0, 1], 'y': 0}),
            equation=FixedValue(0.0)),
        'gamma3': Condition(
            location=CartesianDomain({'x':  1, 'y': [0, 1]}),
            equation=FixedValue(0.0)),
        'gamma4': Condition(
            location=CartesianDomain({'x': 0, 'y': [0, 1]}),
            equation=FixedValue(0.0)),
        'D': Condition(
            location=CartesianDomain({'x': [0, 1], 'y': [0, 1]}),
            equation=Equation(laplace_equation)),
    }

    def poisson_sol(self, pts):
        return -(
            torch.sin(pts.extract(['x'])*torch.pi) *
            torch.sin(pts.extract(['y'])*torch.pi)
        )/(2*torch.pi**2)

    truth_solution = poisson_sol