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FEM1D.jl
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using SparseArrays, QuadGK, LinearAlgebra
function tent(x::Real, center::Real, h::Real)
if center<=x<=(center+h)
return 1 - (x-center)/h
elseif (center-h)<=x<center
return 1 + (x-center)/h
else
return 0
end
end
function d_tent(x::Real, center::Real, h::Real)
if center<=x<=(center+h)
return -1/h
elseif (center-h)<=x<center
return 1/h
else
return 0
end
end
function BVP_DD(l::Integer, a::Function, c::Function, leftBC::Real, rightBC::Real, f::Function)
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
b = Array{Float64,1}(undef, N-2)
x = zeros(N)
for i = 1:(N-2)
xi = h*i
for j = [-1, 0, +1]
if 1<=(i+j)<=N-2
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-10)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-10)[1]
end
b[1] -= leftBC*quadgk(x -> (-a(x)*d_tent(x, h, h)*d_tent(x, 0, h) + c(x)*tent(x, h, h)*tent(x, 0, h)), 0, h, rtol=1e-10)[1]
b[N-2] -= rightBC*quadgk(x -> (-a(x)*d_tent(x, 1-h, h)*d_tent(x, 1, h) + c(x)*tent(x, 1-h, h)*tent(x, 1, h)), 1-h, 1, rtol=1e-10)[1]
A = sparse(rows, columns, values)
x[1], x[N] = leftBC, rightBC
return A, b, x
end
function BVP_DD(l::Integer, a::Function, d::Function, c::Function, leftBC::Real, rightBC::Real, f::Function)
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
b = Array{Float64,1}(undef, N-2)
x = zeros(N)
for i = 1:(N-2)
xi = h*i
for j = [-1, 0, +1]
if 1<=(i+j)<=N-2
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-13)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-13)[1]
end
b[1] -= leftBC*quadgk(x -> (-a(x)*d_tent(x, h, h)*d_tent(x, 0, h) + d(x)*tent(x, h, h)*d_tent(x, 0, h) + c(x)*tent(x, h, h)*tent(x, 0, h)), 0, h, rtol=1e-13)[1]
b[N-2] -= rightBC*quadgk(x -> (-a(x)*d_tent(x, 1-h, h)*d_tent(x, 1, h) + d(x)*tent(x, 1-h, h)*d_tent(x, 1, h) + c(x)*tent(x, 1-h, h)*tent(x, 1, h)), 1-h, 1, rtol=1e-13)[1]
A = sparse(rows, columns, values)
x[1], x[N] = leftBC, rightBC
return A, b, x
end
function BVP_ND(l::Integer, a::Function, d::Function, c::Function, leftBC::Real, rightBC::Real, f::Function)
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
b = Array{Float64,1}(undef, N-1)
x = zeros(N)
xi = 0
i = 1
for j = [0, +1]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi, xi+h, rtol=1e-13)[1])
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi, xi+h, rtol=1e-13)[1]
for i = 2:(N-1)
xi = h*(i-1)
for j = [-1, 0, +1]
if 1<=(i+j)<=N-1
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-13)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-13)[1]
end
b[1] += a(0)*leftBC
b[N-1] -= rightBC*quadgk(x -> (-a(x)*d_tent(x, 1-h, h)*d_tent(x, 1, h) + d(x)*tent(x, 1-h, h)*d_tent(x, 1, h) + c(x)*tent(x, 1-h, h)*tent(x, 1, h)), 1-h, 1, rtol=1e-13)[1]
A = sparse(rows, columns, values)
x[N] = rightBC
return A, b, x
end
function BVP_DN(l::Integer, a::Function, d::Function, c::Function, leftBC::Real, rightBC::Real, f::Function)
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
b = Array{Float64,1}(undef, N-1)
x = zeros(N)
for i = 1:(N-2)
xi = h*i
for j = [-1, 0, +1]
if 1<=(i+j)<=N-1
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-13)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-13)[1]
end
xi = 1
i = N-1
for j = [-1, 0]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi, rtol=1e-13)[1])
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi, rtol=1e-13)[1]
b[1] -= leftBC*quadgk(x -> (-a(x)*d_tent(x, h, h)*d_tent(x, 0, h) + d(x)*tent(x, h, h)*d_tent(x, 0, h) + c(x)*tent(x, h, h)*tent(x, 0, h)), 0, h, rtol=1e-13)[1]
b[N-1] -= a(1)*rightBC
A = sparse(rows, columns, values)
x[1] = leftBC
return A, b, x
end
function BVP_NN(l::Integer, a::Function, d::Function, c::Function, leftBC::Real, rightBC::Real, f::Function)
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
b = Array{Float64,1}(undef, N)
x = zeros(N)
xi = 0
i = 1
for j = [0, +1]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi, xi+h, rtol=1e-13)[1])
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi, xi+h, rtol=1e-13)[1]
for i = 2:(N-1)
xi = h*(i-1)
for j = [-1, 0, +1]
if 1<=(i+j)<=N
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-13)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-13)[1]
end
xi = 1
i = N
for j = [-1, 0]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi, rtol=1e-13)[1])
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi, rtol=1e-13)[1]
b[1] += a(0)*leftBC
b[N] -= a(1)*rightBC
A = sparse(rows, columns, values)
return A, b, x
end
function BVP_1D(l::Integer, F::Array{Function, 1}, BCs::Array{Array{Any,1},1})
N = 2^l+1
h = 1/(N-1)
rows, columns, values = [], [], Array{Float64,1}(undef, 0)
a, d, c, f = F
(nL, tL), (nR, tR) = BCs
DOF = N - 2 + (tL == "N") + (tR == "N")
b = Array{Float64,1}(undef, DOF)
x = zeros(N)
if tL == "N"
first, shift, last = 2, 1, N-1
else
first, shift, last = 1, 0, N-2
end
for i = first:last
xi = h*(i-shift)
for j = [-1, 0, +1]
if 1<=(i+j)<=DOF
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi+h, rtol=1e-13)[1])
end
end
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi+h, rtol=1e-13)[1]
end
if tL == "N"
xi = 0
i = 1
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi, xi+h, rtol=1e-13)[1]
b[i] += a(0)*nL
for j = [0, +1]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi, xi+h, rtol=1e-13)[1])
end
else
i = 1
b[i] -= nL*quadgk(x -> (-a(x)*d_tent(x, h, h)*d_tent(x, 0, h) + d(x)*tent(x, h, h)*d_tent(x, 0, h) + c(x)*tent(x, h, h)*tent(x, 0, h)), 0, h, rtol=1e-13)[1]
x[i] = nL
end
if tR == "N"
xi = 1
i = DOF
b[i] = quadgk(x -> (f(x)*tent(x, xi, h)), xi-h, xi, rtol=1e-13)[1]
b[i] -= a(1)*nR
for j = [-1, 0]
push!(rows, i)
push!(columns, i+j)
push!(values, quadgk(x -> (-a(x)*d_tent(x, xi, h)*d_tent(x, xi+j*h, h) + d(x)*tent(x, xi, h)*d_tent(x, xi+j*h, h) + c(x)*tent(x, xi, h)*tent(x, xi+j*h, h)), xi-h, xi, rtol=1e-13)[1])
end
else
i = DOF
b[i] -= nR*quadgk(x -> (-a(x)*d_tent(x, 1-h, h)*d_tent(x, 1, h) + d(x)*tent(x, 1-h, h)*d_tent(x, 1, h) + c(x)*tent(x, 1-h, h)*tent(x, 1, h)), 1-h, 1, rtol=1e-13)[1]
x[N] = nR
end
A = sparse(rows, columns, values)
return A, b, x
end
function test_1()
errors, H = [], []
for l = [7, 8, 9, 10]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=1:(N-2)]
ex = x->sin(3*pi*x)+2*x+1
a = x->1+x^2
c = x->exp(-(x-1/2)^2)
leftBC, rightBC = 1, 3
f = x->(2*x*(3*pi*cos(3*pi*x)+2)-(3*pi)^2*(1+x^2)*sin(3*pi*x))+exp(-(x-1/2)^2)*(sin(3*pi*x)+2*x+1)
A, b, x = BVP_DD(l, a, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
end
function test_2()
errors, H = [], []
for l = [7, 8, 9, 10]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=1:(N-2)]
ex = x->exp(-x)
a = x->exp(x)
c = x->sin(pi*x)^2+1
leftBC, rightBC = 1, exp(-1)
f = x->exp(-x)*(sin(pi*x)^2+1)
A, b, x= BVP_DD(l, a, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
end
function test_3()
errors, H = [], []
for l = [7, 8, 9, 10, 11, 12]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=1:(N-2)]
ex = x -> exp(-x)
a = x -> exp(x)
d = x -> x^2
c = x -> sin(pi*x)^2+1
leftBC, rightBC = 1, exp(-1)
f = x -> exp(-x)*(sin(pi*x)^2+1) - exp(-x)*x^2
A, b, x= BVP_DD(l, a, d, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
return errors, H, slope
end
function test_4()
errors, H = [], []
for l = [7, 8, 9, 10, 11, 12]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=0:(N-2)]
ex = x -> x^2 + 1
a = x -> (x + 1)
d = x -> cos(pi*x)^2
c = x -> sin(pi*x)^2+1
leftBC, rightBC = 0, 2
f = x -> (x^2+1)*(sin(pi*x)^2+1) + 2*x*cos(pi*x)^2 + (4*x+2)
A, b, x= BVP_ND(l, a, d, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
return errors, H, slope
end
function test_5()
errors, H = [], []
for l = [7, 8, 9, 10, 11, 12]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=1:(N-1)]
ex = x -> x^2 + 1
a = x -> (x + 1)
d = x -> cos(pi*x)^2
c = x -> sin(pi*x)^2+1
leftBC, rightBC = 1, 2
f = x -> (x^2+1)*(sin(pi*x)^2+1) + 2*x*cos(pi*x)^2 + (4*x+2)
A, b, x= BVP_DN(l, a, d, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
return errors, H, slope
end
function test_6()
errors, H = [], []
for l = [7, 8, 9, 10, 11, 12]
N = 2^l+1
h = 1/(N-1)
coords = [h*i for i=0:(N-1)]
ex = x -> x^2 + 1
a = x -> (x + 1)
d = x -> cos(pi*x)^2
c = x -> sin(pi*x)^2+1
leftBC, rightBC = 0, 2
f = x -> (x^2+1)*(sin(pi*x)^2+1) + 2*x*cos(pi*x)^2 + (4*x+2)
A, b, x= BVP_NN(l, a, d, c, leftBC, rightBC, f)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
return errors, H, slope
end
function test(tL::String, tR::String)
errors, H = [], []
if tL == "N"
leftBC = 1
start = 0
elseif tL == "D"
leftBC = 1
start = 1
end
if tR == "N"
rightBC = 3
elseif tR == "D"
rightBC = 3
end
for l = [7, 8, 9, 10, 11, 12]
N = 2^l+1
DOF = N - 2 + (tL == "N") + (tR == "N")
h = 1/(N-1)
coords = [h*i for i=start:(DOF-1+start)]
ex = x -> x^2 + x + 1
a = x -> (x + 1)
d = x -> cos(pi*x)^2
c = x -> sin(pi*x)^2 + 1
f = x -> (x^2 + x + 1)*(sin(pi*x)^2 + 1) + (2*x + 1)*cos(pi*x)^2 + (4*x + 3)
F = [a, d, c, f]
BCs = [[leftBC, tL], [rightBC, tR]]
A, b, x = BVP_1D(l, F, BCs)
error = norm(ex.(coords) - inv(Array(A))*b, Inf)
push!(H, h)
push!(errors, error)
end
slope = log(errors[1]/errors[end])/log(H[1]/H[end])
return errors, H, slope
end