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H1error_test.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Fri Oct 11 11:38:59 2019
@author: maged
"""
import numpy as np
import matplotlib.pyplot as plt
import scipy.linalg as la
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from quad import *
from funs import *
element_linear_num = 32
node_linear_num = element_linear_num + 1
element_num = element_linear_num * element_linear_num
node_num = node_linear_num * node_linear_num
a = 0
b = 1.0
grid = np.linspace ( a, b, node_linear_num )
h1_error = 0.0
#
quad_num, quad_point, quad_weight = quad()
#
# x and y for each node.
#
x = np.zeros( node_linear_num * node_linear_num)
y = np.zeros( node_linear_num * node_linear_num)
v = 0
for j in range (0, node_linear_num):
for i in range (0, node_linear_num):
x[v]= grid[i]
y[v] = grid[j]
v = v + 1
#
#
# Memory allocation.
#
A = np.zeros((node_num, node_num))
rhs = np.zeros(node_num)
for ex in range ( 0, element_linear_num ):
w = ex
e = ex + 1
xw = grid[w]
xe = grid[e]
for ey in range ( 0, element_linear_num ):
s = ey
n = ey + 1
ys = grid[s]
yn = grid[n]
sw = ey * node_linear_num + ex
se = ey * node_linear_num + ex + 1
nw = ( ey + 1 ) * node_linear_num + ex
ne = ( ey + 1 ) * node_linear_num + ex + 1
#
# The 2D quadrature rule is the 'product' of X and Y copies of the 1D rule.
#
for qx in range (0, quad_num):
xq = xw + quad_point[qx] * (xe - xw)
for qy in range(0,quad_num):
yq = ys + quad_point[qy] * (yn - ys)
wq = quad_weight[qx] * quad_weight[qy] * (xe - xw) * (yn - ys)
#
# Evaluate all four basis functions, and their X and Y derivatives.
#
vsw = ( xe - xq ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vswx = ( -1.0 ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vswy = ( xe - xq ) / ( xe - xw ) * ( -1.0 ) / ( yn - ys )
vse = ( xq - xw ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vsex = ( 1.0 ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vsey = ( xq - xw ) / ( xe - xw ) * ( -1.0 ) / ( yn - ys )
vnw = ( xe - xq ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vnwx = ( -1.0 ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vnwy = ( xe - xq ) / ( xe - xw ) * ( 1.0 ) / ( yn - ys )
vne = ( xq - xw ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vnex = ( 1.0 ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vney = ( xq - xw ) / ( xe - xw ) * ( 1.0 ) / ( yn - ys )
#
# Compute contributions to the stiffness matrix.
#
A[sw,sw] = A[sw,sw] + wq * ( vswx * vswx + vswy * vswy )
A[sw,se] = A[sw,se] + wq * ( vswx * vsex + vswy * vsey )
A[sw,nw] = A[sw,nw] + wq * ( vswx * vnwx + vswy * vnwy )
A[sw,ne] = A[sw,ne] + wq * ( vswx * vnex + vswy * vney )
rhs[sw] = rhs[sw] + wq * vsw * rhs_fn ( xq, yq )
A[se,sw] = A[se,sw] + wq * ( vsex * vswx + vsey * vswy )
A[se,se] = A[se,se] + wq * ( vsex * vsex + vsey * vsey )
A[se,nw] = A[se,nw] + wq * ( vsex * vnwx + vsey * vnwy )
A[se,ne] = A[se,ne] + wq * ( vsex * vnex + vsey * vney )
rhs[se] = rhs[se] + wq * vse * rhs_fn ( xq, yq )
A[nw,sw] = A[nw,sw] + wq * ( vnwx * vswx + vnwy * vswy )
A[nw,se] = A[nw,se] + wq * ( vnwx * vsex + vnwy * vsey )
A[nw,nw] = A[nw,nw] + wq * ( vnwx * vnwx + vnwy * vnwy )
A[nw,ne] = A[nw,ne] + wq * ( vnwx * vnex + vnwy * vney )
rhs[nw] = rhs[nw] + wq * vnw * rhs_fn ( xq, yq )
A[ne,sw] = A[ne,sw] + wq * ( vnex * vswx + vney * vswy )
A[ne,se] = A[ne,se] + wq * ( vnex * vsex + vney * vsey )
A[ne,nw] = A[ne,nw] + wq * ( vnex * vnwx + vney * vnwy )
A[ne,ne] = A[ne,ne] + wq * ( vnex * vnex + vney * vney )
rhs[ne] = rhs[ne] + wq * vne * rhs_fn( xq, yq )
A_in = A
# Modify the linear system to enforce the boundary conditions where
# X = 0 or 1 or Y = 0 or 1.
#
v = 0
for j in range ( 0, node_linear_num ):
for i in range ( 0, node_linear_num ):
if ( i == 0 or i == node_linear_num - 1 or j == 0 or j == node_linear_num - 1 ):
A[v,0:node_num] = 0.0
A[v,v] = 1.0
rhs[v] = 0.0
v = v + 1
#
# Solve the linear system.
#
u = la.solve(A, rhs)
# u_mat is the solution u in each matrix
u_mat = np.zeros((node_linear_num,node_linear_num))
v = 0
for j in range ( 0, node_linear_num ):
for i in range ( 0, node_linear_num ):
u_mat[i,j] = u[v]
v = v + 1
print(u_mat)
u_exact = exact_fn(x, y)
#
# Compare the solution and the error at the nodes.
#
print ( '' )
print ( ' Node x y u u_exact' )
print ( '' )
v = 0
for j in range ( 0, node_linear_num ):
for i in range ( 0, node_linear_num ):
print ( ' %4d %8f %8f %14g %14g' % ( v, x[v], y[v], u[v], u_exact[v] ) )
v = v + 1
# Quadrature defination
quad_num = 3
quad_point = np.array (( \
-0.774596669241483377035853079956, \
0.0, \
0.774596669241483377035853079956 ) )
quad_weight = np.array (( \
5.0 / 9.0, \
8.0 / 9.0, \
5.0 / 9.0 ))
#
# x and y for each node.
#
x = np.zeros( node_linear_num * node_linear_num)
y = np.zeros( node_linear_num * node_linear_num)
v = 0
for j in range (0, node_linear_num):
for i in range (0, node_linear_num):
x[v]= grid[i]
y[v] = grid[j]
v = v + 1
for ex in range ( 0, element_linear_num ):
w = ex
e = ex + 1
xw = grid[w]
xe = grid[e]
for ey in range ( 0, element_linear_num ):
s = ey
n = ey + 1
ys = grid[s]
yn = grid[n]
sw = ey * node_linear_num + ex
se = ey * node_linear_num + ex + 1
nw = ( ey + 1 ) * node_linear_num + ex
ne = ( ey + 1 ) * node_linear_num + ex + 1
#
# The 2D quadrature rule is the 'product' of X and Y copies of the 1D rule.
#
for qx in range (0, quad_num):
xq = (( 1.0 - quad_point[qx] ) * xw
+ ( 1.0 + quad_point[qx] ) * xe ) / 2.0
for qy in range(0,quad_num):
yq = (( 1.0 - quad_point[qy] ) * ys
+ ( 1.0 + quad_point[qy] ) * yn ) / 2.0
wq = quad_weight[qx] * quad_weight[qy] * (xe - xw) / 2.0 * (yn - ys) / 2.0
#
# Evaluate all four basis functions, and their X and Y derivatives.
#
vswx = ( -1.0 ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vswy = ( xe - xq ) / ( xe - xw ) * ( -1.0 ) / ( yn - ys )
vsex = ( 1.0 ) / ( xe - xw ) * ( yn - yq ) / ( yn - ys )
vsey = ( xq - xw ) / ( xe - xw ) * ( -1.0 ) / ( yn - ys )
vnwx = ( -1.0 ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vnwy = ( xe - xq ) / ( xe - xw ) * ( 1.0 ) / ( yn - ys )
vnex = ( 1.0 ) / ( xe - xw ) * ( yq - ys ) / ( yn - ys )
vney = ( xq - xw ) / ( xe - xw ) * ( 1.0 ) / ( yn - ys )
uxq = u_mat[w,s] * vswx + u_mat[e,s] * vsex + u_mat[w,n] * vnwx + u_mat[e,n] * vnex
uyq = u_mat[w,s] * vswy + u_mat[e,s] * vsey + u_mat[w,n] * vnwy + u_mat[e,n] * vney
exq = exact_fnx(xq, yq)
eyq = exact_fny(xq, yq)
h1_error = h1_error + wq * ( ( uxq - exq ) * ( uxq - exq ) + ( uyq - eyq ) * (uyq - eyq) )
h1_error = np.sqrt(h1_error)
print('\n The H1 error = ', h1_error)