Poisson Solvers
Solve
\[- \Delta \phi(x) = f(x)\]
in one dimension, with a spectral or a matrix-free finite-difference solver on a uniform periodic grid, or with a B-spline Galerkin solver. A Potential pairs the solution with the solver that produced it, so that it can be re-solved in place as the source changes.
Evaluate a potential with p(x), and its d-th derivative with p(x, d).
using Plots
using PoissonSolvers
sol(x) = sin(2π * x)
der(x) = 2π * cos(2π * x)
rhs(x) = 4π^2 * sin(2π * x)
domain = (0.0, 1.0)
x = LinRange(domain[begin], domain[end], 200)200-element LinRange{Float64, Int64}:
0.0, 0.00502513, 0.0100503, 0.0150754, …, 0.984925, 0.98995, 0.994975, 1.0FFT Solver
The grid solver is spectral in the coefficients, but evaluates at the nearest grid point, so the plotted curve is a staircase of width Δx.
b = FFTWBasis(domain, 64)
p = Potential(b, rhs)Potential{PoissonSolvers.FFTWSolution{Vector{Float64}, FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, PoissonSolverFFT{Float64, FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}, Vector{Float64}}(PoissonSolvers.FFTWSolution{Vector{Float64}, FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}(FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}((0.0, 1.0), 0.0:0.015625:1.0, 0.015625), [-1.3392341810760245e-16, 0.09801714032956048, 0.19509032201612814, 0.2902846772544623, 0.38268343236508967, 0.47139673682599753, 0.5555702330196021, 0.6343932841636454, 0.7071067811865475, 0.7730104533627369 … -0.8314696123025453, -0.7730104533627371, -0.7071067811865477, -0.6343932841636456, -0.5555702330196023, -0.47139673682599775, -0.3826834323650899, -0.2902846772544625, -0.19509032201612841, -0.09801714032956074]), PoissonSolverFFT{Float64, FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}(FFTWBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}((0.0, 1.0), 0.0:0.015625:1.0, 0.015625), FFTW real-to-complex plan for 64-element array of Float64
(rdft2-r2hc-direct-64 "r2cf_64"), 0.015625 * FFTW complex-to-real plan for 33-element array of ComplexF64
(rdft2-hc2r-direct-64 "r2cb_64"), [0.0, 0.025330295910584444, 0.006332573977646111, 0.0028144773233982714, 0.0015831434944115277, 0.0010132118364233778, 0.0007036193308495678, 0.0005169448145017233, 0.00039578587360288194, 0.00031271970259980796 … 4.788335710885529e-5, 4.397620817809799e-5, 4.052847345693511e-5, 3.747085193873438e-5, 3.4746633622200886e-5, 3.230905090635771e-5, 3.0119257919838814e-5, 2.8144773233982723e-5, 2.6358268377299114e-5, 2.473661710018012e-5], ComplexF64[0.0 + 0.0im, -4.299339575620032e-15 - 32.0im, -6.125206681782853e-17 - 3.854178191371115e-16im, -6.087436351144207e-17 - 2.026477856188795e-16im, 2.2213370792295974e-18 - 7.889752062502751e-17im, 6.166910423976222e-17 - 5.0081560096837867e-17im, 3.201180578720749e-17 + 5.3981048515317e-19im, 1.184932856688489e-17 + 1.881940566350671e-17im, 4.733055561345467e-18 + 1.3578636865187278e-17im, 7.322116512088635e-18 - 2.2165481148044987e-17im … 1.3369965518648538e-18 + 6.812993106208358e-18im, -4.1314023200665753e-19 + 2.5885841877639575e-19im, 1.1515928218213234e-18 + 8.994971558935005e-19im, -7.867876576046666e-19 + 1.3685364579246387e-18im, -1.5941950924830796e-18 + 4.823478967095653e-19im, 9.082052476721393e-20 - 3.166182686316005e-21im, -8.159711654263059e-19 + 1.243640171080903e-18im, -1.2652387428845249e-18 - 5.591680267249299e-22im, -1.662229887809953e-19 + 0.0im, -2.978456462313027e-19 + 0.0im]), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])plot(xlabel = "x", ylabel = "ϕ(x)")
plot!(x, p.(x); xlims = domain, label = "Solution")
plot!(x, sol.(x); xlims = domain, label = "Reference")
Matrix-Free Solver
The matrix-free solver uses the same grid, and applies a central difference stencil of order 2 or 4 by conjugate gradients rather than a transform. Its solution evaluates like the FFT solver's.
b = FiniteDifferenceBasis(domain, 64; order = 4)
p = Potential(b, rhs)Potential{PoissonSolvers.FFTWSolution{Vector{Float64}, FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, PoissonSolverMatrixFree{Float64, FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}, Vector{Float64}}(PoissonSolvers.FFTWSolution{Vector{Float64}, FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}(FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}((0.0, 1.0), 0.0:0.015625:1.0, 0.015625, 4), [5.534339187631598e-16, 0.09801724141435114, 0.19509052321220693, 0.2902849766242009, 0.38268382702539405, 0.4713972229760731, 0.5555708059775578, 0.634393938411583, 0.7071075104237018, 0.7730112505661556 … -0.831470469794724, -0.7730112505661557, -0.7071075104237018, -0.634393938411583, -0.5555708059775574, -0.4713972229760724, -0.3826838270253931, -0.29028497662419994, -0.19509052321220582, -0.09801724141435005]), PoissonSolverMatrixFree{Float64, FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}(FiniteDifferenceBasis{Float64, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}((0.0, 1.0), 0.0:0.015625:1.0, 0.015625, 4), 3.552713678800501e-15, 256, [-1.6205753732802154e-14, -1.0945970657217665e-13, 1.3220927123571167e-14, -4.8211937003661654e-15, -9.22789840005559e-15, 8.998832519672784e-14, 1.2773694911469478e-13, -2.9876212057490285e-14, 7.68432224527302e-14, 1.890536380844009e-13 … 4.3707090843460326e-14, 1.3446421429821564e-14, 1.4656786392803668e-13, -8.273714947010565e-14, -4.23492678315767e-14, -2.0053696884838586e-14, -2.302893653648212e-15, -6.634463284690261e-16, 1.0386316167787125e-13, 2.8903025319164647e-14], [2.620348391752036e-13, 2.5763864253471676e-13, 2.3760437004834834e-13, 6.718434912874826e-14, 2.5048174552262157e-14, 9.11774703954877e-14, 5.756128618778357e-14, -7.684969130641561e-14, -9.271437460112881e-14, -1.6150080475198483e-13 … 6.485245830757924e-14, 2.973186913412758e-14, 1.595789535063805e-13, 9.420052062153321e-14, 1.1000599644891155e-13, 1.2239659031448672e-13, 2.3736811560276215e-13, 2.957375217596287e-13, 3.470831419862576e-13, 2.7245631457641814e-13], [-4.5728295190576835e-10, 1.5724535836843757e-9, 8.7300583932411e-12, -7.704391716108051e-10, 5.389027384567584e-12, 3.0382337396924795e-10, -7.215889897708753e-10, 9.938086832619874e-10, 4.5125246621380596e-10, -1.8423521013916476e-9 … -4.5045294111082963e-10, 1.0213815502543139e-10, -1.2095024901624764e-9, 1.34971544514279e-9, -1.3731127366516599e-10, -7.299962321671569e-10, 2.6984502470309073e-10, 7.424268033793802e-10, -3.946760128891013e-10, -1.7865303770699146e-10]), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])plot(xlabel = "x", ylabel = "ϕ(x)")
plot!(x, p.(x); xlims = domain, label = "Solution")
plot!(x, sol.(x); xlims = domain, label = "Reference")
Periodic B-Spline Solver
A spline of order $k$ converges at order $k$, and evaluates wherever it is asked.
b = PeriodicBasisSpline(domain, 5, 32)
p = Potential(b, rhs)Potential{SimpleSplines.Spline{Float64, SimpleSplines.PeriodicBSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, Vector{Float64}}, PoissonSolverSpline{Float64, SimpleSplines.SplineQuadrature{Float64, SimpleSplines.PeriodicBSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}}, SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}}, Vector{Float64}}(Spline(PeriodicBSplineBasis, (32,)), PoissonSolverSpline{Float64, SimpleSplines.SplineQuadrature{Float64, SimpleSplines.PeriodicBSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}}, SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}}(SimpleSplines.SplineQuadrature{Float64, SimpleSplines.PeriodicBSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}}(SimpleSplines.PeriodicBSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}(SimpleSplines.UniformMesh{Float64}(32, 0.0, 1.0), 4, [-2.0, -1.96875, -1.9375, -1.90625, -1.875, -1.84375, -1.8125, -1.78125, -1.75, -1.71875 … 2.6875, 2.71875, 2.75, 2.78125, 2.8125, 2.84375, 2.875, 2.90625, 2.9375, 2.96875], [0.0, 0.03125, 0.0625, 0.09375, 0.125, 0.15625, 0.1875, 0.21875, 0.25, 0.28125 … 0.71875, 0.75, 0.78125, 0.8125, 0.84375, 0.875, 0.90625, 0.9375, 0.96875, 1.0], 64), 6, [0.0010551638405757497, 0.005293603336464617, 0.011896575217450049, 0.019353424782549953, 0.025956396663535383, 0.03019483615942425, 0.03230516384057575, 0.036543603336464614, 0.04314657521745005, 0.05060342478254995 … 0.94939657521745, 0.95685342478255, 0.9634563966635353, 0.9676948361594242, 0.9698051638405758, 0.9740436033364647, 0.98064657521745, 0.98810342478255, 0.9947063966635353, 0.9989448361594242], [0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537, 0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975 … 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537, 0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537], SparseArrays.SparseMatrixCSC{Float64, Int64}[sparse([1, 29, 30, 31, 32, 1, 29, 30, 31, 32 … 28, 29, 30, 31, 32, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 191, 191, 191, 191, 191, 192, 192, 192, 192, 192], [5.415870500882965e-8, 0.036317787337507235, 0.44118472007767584, 0.47491200909240605, 0.04758542933370592, 3.4307909281240866e-5, 0.01983198937695191, 0.3687551355036508, 0.533632762004258, 0.077745805205858 … 3.4307909281242235e-5, 0.07774580520585893, 0.5336327620042594, 0.36875513550364886, 0.019831989376951572, 5.4158705008840255e-8, 0.04758542933370592, 0.47491200909240605, 0.44118472007767584, 0.036317787337507235], 32, 192), sparse([1, 29, 30, 31, 32, 1, 29, 30, 31, 32 … 28, 29, 30, 31, 32, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 191, 191, 191, 191, 191, 192, 192, 192, 192, 192], [0.00020530917730948012, -4.811125603829049, -16.486340724940547, 15.406263570545544, 5.890997449046741, 0.02592404991504613, -3.0562006944226834, -17.436672150107285, 12.067870433397669, 8.399078361217255 … -0.025924049915046907, -8.399078361217326, -12.067870433397566, 17.43667215010729, 3.0562006944226443, -0.0002053091773095103, -5.890997449046741, -15.406263570545544, 16.486340724940547, 4.811125603829049], 32, 192), sparse([1, 29, 30, 31, 32, 1, 29, 30, 31, 32 … 28, 29, 30, 31, 32, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 191, 191, 191, 191, 191, 192, 192, 192, 192, 192], [0.5837269135306606, 478.0081181855429, -410.6080814701594, -612.2244647027792, 544.2407010738649, 14.691722216776306, 353.23092808750533, -50.38450647929193, -944.2320490871562, 626.6939052621668 … 14.6917222167766, 626.6939052621678, -944.2320490871632, -50.38450647928357, 353.2309280875023, 0.5837269135307178, 544.2407010738649, -612.2244647027792, -410.6080814701594, 478.0081181855429], 32, 192), sparse([1, 29, 30, 31, 32, 1, 29, 30, 31, 32 … 28, 29, 30, 31, 32, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 191, 191, 191, 191, 191, 192, 192, 192, 192, 192], [1106.4194792955573, -31661.58052070439, 93878.32208281755, -91665.48312422635, 28342.322082817558, 5550.745412136722, -27217.25458786334, 76101.01835145336, -64999.527527180035, 10565.018351453356 … -5550.745412136777, -10565.018351452889, 64999.527527179336, -76101.01835145289, 27217.254587863223, -1106.4194792956114, -28342.322082817558, 91665.48312422635, -93878.32208281755, 31661.58052070439], 32, 192)], SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}(sparse([1, 2, 3, 4, 5, 29, 30, 31, 32, 1 … 32, 1, 2, 3, 4, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 1, 1, 1, 1, 2 … 31, 32, 32, 32, 32, 32, 32, 32, 32, 32], [0.013450555279982358, 0.00759841407627866, 0.001257991622574956, 4.323054453262787e-5, 8.611662257495591e-8, 8.611662257495565e-8, 4.3230544532627896e-5, 0.0012579916225749558, 0.007598414076278658, 0.00759841407627866 … 0.007598414076278657, 0.007598414076278658, 0.0012579916225749558, 4.3230544532627896e-5, 8.611662257495565e-8, 8.611662257495589e-8, 4.323054453262787e-5, 0.0012579916225749547, 0.007598414076278657, 0.013450555279982358], 32, 32), ComplexF64[32.00000000000001 - 0.0im, 32.51836075515856 + 5.734393464784878e-16im, 34.12649005637539 + 6.839505320875802e-16im, 36.993067779212176 + 1.4839727866859543e-15im, 41.43355790737018 + 2.478071156717412e-15im, 47.97149192457595 + 4.490633117312843e-15im, 57.44755505556277 + 6.2301190951086196e-15im, 71.21020769974197 + 9.345436589313703e-15im, 91.45161290322585 + 1.450818198668856e-14im, 121.81020559441068 + 2.413341962133079e-14im, 168.46952253901662 + 4.489060937840186e-14im, 242.15990394489913 + 8.902179143905087e-14im, 361.5668823462916 + 1.320110250767593e-13im, 557.2814232480357 + 2.0196944953617203e-13im, 864.679119891131 + 2.0941251433807327e-13im, 1252.4620859555266 + 5.099301835318973e-13im, 1463.2258064516263 - 0.0im], FFTW real-to-complex plan for 32-element array of Float64
(rdft2-r2hc-direct-32 "r2cf_32"), 0.03125 * FFTW complex-to-real plan for 17-element array of ComplexF64
(rdft2-hc2r-direct-32 "r2cb_32"), ComplexF64[1.0e-323 + 1.5e-323im, 3.5e-323 + 4.0e-323im, 6.0e-323 + 9.4e-323im, 1.0e-322 + 1.1e-322im, 1.2e-322 + 1.73e-322im, 1.8e-322 + 1.9e-322im, 1.93e-322 + 2.03e-322im, 2.08e-322 + 2.27e-322im, 2.3e-322 + 2.4e-322im, 3.16e-322 + 3.2e-322im, 3.26e-322 + 3.8e-322im, 3.85e-322 + 3.9e-322im, 3.95e-322 + 4.3e-322im, 4.35e-322 + 4.4e-322im, 4.45e-322 + 4.6e-322im, 4.7e-322 + 4.74e-322im, 5.43e-322 + 5.7e-322im], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 32), [0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997 … 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125000000000001, 0.031249999999999993, 0.03125], [6.93188582921237e-310, 6.93188582921237e-310, 6.9318065556562e-310, 6.93188582921237e-310, 6.9318058706986e-310, 6.93188582921237e-310, 6.93188582921237e-310, 6.93188582921237e-310, 6.93188582921237e-310, 6.93188582921237e-310 … 6.93188582921237e-310, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Dict{Tuple{Int64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}((1, 1) => sparse([1, 2, 3, 4, 5, 29, 30, 31, 32, 1 … 32, 1, 2, 3, 4, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 1, 1, 1, 1, 2 … 31, 32, 32, 32, 32, 32, 32, 32, 32, 32], [15.555555555555554, -0.9777777777777784, -6.044444444444443, -0.749206349206349, -0.006349206349206349, -0.00634920634920633, -0.7492063492063495, -6.044444444444442, -0.9777777777777824, -0.9777777777777784 … -0.9777777777777897, -0.9777777777777829, -6.044444444444442, -0.7492063492063494, -0.00634920634920633, -0.006349206349206357, -0.7492063492063492, -6.044444444444435, -0.9777777777777901, 15.555555555555568], 32, 32))), SimpleSplines.CirculantMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, FFTW.rFFTWPlan{Float64, -1, false, 1, Tuple{Int64}}, AbstractFFTs.ScaledPlan{ComplexF64, FFTW.rFFTWPlan{ComplexF64, 1, false, 1, UnitRange{Int64}}, Float64}}(sparse([1, 2, 3, 4, 5, 29, 30, 31, 32, 1 … 32, 1, 2, 3, 4, 28, 29, 30, 31, 32], [1, 1, 1, 1, 1, 1, 1, 1, 1, 2 … 31, 32, 32, 32, 32, 32, 32, 32, 32, 32], [15.555555555555554, -0.9777777777777784, -6.044444444444443, -0.749206349206349, -0.006349206349206349, -0.00634920634920633, -0.7492063492063495, -6.044444444444442, -0.9777777777777824, -0.9777777777777784 … -0.9777777777777897, -0.9777777777777829, -6.044444444444442, -0.7492063492063494, -0.00634920634920633, -0.006349206349206357, -0.7492063492063492, -6.044444444444435, -0.9777777777777901, 15.555555555555568], 32, 32), ComplexF64[0.0 + 0.0im, 0.8236997004537485 + 1.791616269226234e-16im, 0.21610852276415743 + 3.1185127145172414e-17im, 0.10411614873002797 + 1.1587867541049512e-17im, 0.06559525496374129 + 6.2059789617957516e-18im, 0.04860521303066781 + 4.228460661636128e-18im, 0.04042088656052677 + 3.7835373821174274e-18im, 0.03681042528851937 + 3.628748110761029e-18im, 0.03619025735294119 + 4.653112481179681e-18im, 0.03807451118985645 + 6.4182606036709284e-18im, 0.04261176588610447 + 8.696968413748671e-18im, 0.05048572436609807 + 1.3548388487679715e-17im, 0.0629267108143193 + 1.977923946787117e-17im, 0.08148646700745453 + 3.086740205074813e-17im, 0.10652422857894013 + 3.651649393861571e-17im, 0.1325949714126268 + 3.634793909272525e-17im, 0.14476102941176455 - 0.0im], FFTW real-to-complex plan for 32-element array of Float64
(rdft2-r2hc-direct-32 "r2cf_32"), 0.03125 * FFTW complex-to-real plan for 17-element array of ComplexF64
(rdft2-hc2r-direct-32 "r2cb_32"), ComplexF64[-0.0 + 0.0im, 7.603190775705967 - 14.224569451076755im, 1.857673808859708e-16 - 8.173281376769962e-17im, -1.0970584048100913e-16 - 7.967340067841367e-17im, 4.524137706702832e-17 + 4.6307879806773255e-17im, 1.1384306939964005e-18 + 3.268726706551422e-17im, 6.84402932224854e-17 + 1.516321036212229e-17im, -3.2229757625055455e-33 + 3.269422536124666e-17im, -6.02688854705178e-18 + 2.1094109914681218e-17im, 5.700560560192017e-33 - 3.3816959179461435e-17im, 3.8619410138724655e-17 - 2.38213541511312e-17im, -2.3602641449512716e-17 - 5.57287407215446e-17im, -8.469527148579564e-18 + 2.5063741355968145e-18im, 4.967397653407105e-17 + 8.239268410398743e-17im, 1.51890673208569e-16 + 2.577292263935162e-17im, 6.456691020546624e-32 - 2.355359843389003e-16im, -6.83047368665867e-17 + 0.0im], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 32)), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])plot(xlabel = "x", ylabel = "ϕ(x)")
plot!(x, p.(x); xlims = domain, label = "Solution")
plot!(x, sol.(x); xlims = domain, label = "Reference")
plot(xlabel = "x", ylabel = "ϕ'(x)")
plot!(x, p.(x, 1); xlims = domain, label = "Derivative")
plot!(x, der.(x); xlims = domain, label = "Reference")
Dirichlet B-Spline Solver
A periodic basis needs a periodic problem. For a source that does not repeat, use a basis recombined to vanish at both ends, which solves the same equation with $\phi(a) = \phi(b) = 0$.
dsol(x) = sin(π * x)
drhs(x) = π^2 * sin(π * x)
b = DirichletBasisSpline(domain, 5, 32)
p = Potential(b, drhs)Potential{SimpleSplines.Spline{Float64, SimpleSplines.RecombinedBSplineBasis{Float64, SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.Dirichlet, SimpleSplines.Dirichlet}, Vector{Float64}}, PoissonSolverSpline{Float64, SimpleSplines.SplineQuadrature{Float64, SimpleSplines.RecombinedBSplineBasis{Float64, SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.Dirichlet, SimpleSplines.Dirichlet}, SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}}, SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}}, Vector{Float64}}(Spline(RecombinedBSplineBasis, (34,)), PoissonSolverSpline{Float64, SimpleSplines.SplineQuadrature{Float64, SimpleSplines.RecombinedBSplineBasis{Float64, SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.Dirichlet, SimpleSplines.Dirichlet}, SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}}, SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}}(SimpleSplines.SplineQuadrature{Float64, SimpleSplines.RecombinedBSplineBasis{Float64, SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.Dirichlet, SimpleSplines.Dirichlet}, SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}}(SimpleSplines.RecombinedBSplineBasis{Float64, SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}, SimpleSplines.Dirichlet, SimpleSplines.Dirichlet}(SimpleSplines.BSplineBasis{Float64, SimpleSplines.UniformMesh{Float64}}(SimpleSplines.UniformMesh{Float64}(32, 0.0, 1.0), 4, [0.0, 0.0, 0.0, 0.0, 0.0, 0.03125, 0.0625, 0.09375, 0.125, 0.15625 … 0.84375, 0.875, 0.90625, 0.9375, 0.96875, 1.0, 1.0, 1.0, 1.0, 1.0], [0.0, 0.03125, 0.0625, 0.09375, 0.125, 0.15625, 0.1875, 0.21875, 0.25, 0.28125 … 0.71875, 0.75, 0.78125, 0.8125, 0.84375, 0.875, 0.90625, 0.9375, 0.96875, 1.0]), Dirichlet(), Dirichlet(), sparse([2, 3, 4, 5, 6, 7, 8, 9, 10, 11 … 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10 … 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 … 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0], 36, 34), [1, 1, 2, 3, 4, 5, 6, 7, 8, 9 … 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13 … 26, 27, 28, 29, 30, 31, 32, 33, 34, 34], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11 … 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], 5), 6, [0.0010551638405757497, 0.005293603336464617, 0.011896575217450049, 0.019353424782549953, 0.025956396663535383, 0.03019483615942425, 0.03230516384057575, 0.036543603336464614, 0.04314657521745005, 0.05060342478254995 … 0.94939657521745, 0.95685342478255, 0.9634563966635353, 0.9676948361594242, 0.9698051638405758, 0.9740436033364647, 0.98064657521745, 0.98810342478255, 0.9947063966635353, 0.9989448361594242], [0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537, 0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975 … 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537, 0.002676945193424537, 0.005636899578877166, 0.0073111552276982975, 0.0073111552276982975, 0.005636899578877166, 0.002676945193424537], SparseArrays.SparseMatrixCSC{Float64, Int64}[sparse([1, 2, 3, 4, 1, 2, 3, 4, 1, 2 … 33, 34, 31, 32, 33, 34, 31, 32, 33, 34], [1, 1, 1, 1, 2, 2, 2, 2, 3, 3 … 190, 190, 191, 191, 191, 191, 192, 192, 192, 192], [0.1250671780952831, 0.003280659321210364, 2.5212324621944765e-5, 5.415870500882965e-8, 0.45180975707154214, 0.06923358297696461, 0.002954606995370426, 3.4307909281240866e-5, 0.5652551450523896, 0.25727516219360563 … 0.2572751621936049, 0.56525514505239, 3.4307909281242235e-5, 0.002954606995370511, 0.06923358297696583, 0.45180975707154464, 5.4158705008840255e-8, 2.521232462194845e-5, 0.003280659321210678, 0.12506717809528875], 34, 192), sparse([1, 2, 3, 4, 1, 2, 3, 4, 1, 2 … 33, 34, 31, 32, 33, 34, 31, 32, 33, 34], [1, 1, 1, 1, 2, 2, 2, 2, 3, 3 … 190, 190, 191, 191, 191, 191, 192, 192, 192, 192], [109.30812275153785, 6.087431476802178, 0.07125495438042025, 0.00020530917730948012, 48.544606465626, 23.157854622797867, 1.6204315278049872, 0.02592404991504613, -7.129140173828837, 30.415938693150743 … -30.41593869315076, 7.129140173828699, -0.025924049915046907, -1.6204315278050174, -23.15785462279801, -48.54460646562539, -0.0002053091773095103, -0.07125495438042713, -6.087431476802457, -109.30812275153697], 34, 192), sparse([1, 2, 3, 4, 1, 2, 3, 4, 1, 2 … 33, 34, 31, 32, 33, 34, 31, 32, 33, 34], [1, 1, 1, 1, 2, 2, 2, 2, 3, 3 … 190, 190, 191, 191, 191, 191, 192, 192, 192, 192], [-17006.09214453346, 5399.875537201007, 133.4380439658558, 0.5837269135306606, -11807.774146325486, 2744.127991964666, 571.4121580439544, 14.691722216776306, -5398.343261824598, -329.81212377531176 … -329.812123775302, -5398.343261824621, 14.6917222167766, 571.4121580439587, 2744.127991964636, -11807.774146325426, 0.5837269135307178, 133.4380439658621, 5399.875537200971, -17006.09214453339], 34, 192), sparse([1, 2, 3, 4, 1, 2, 3, 4, 1, 2 … 33, 34, 31, 32, 33, 34, 31, 32, 33, 34], [1, 1, 1, 1, 2, 2, 2, 2, 3, 3 … 190, 190, 191, 191, 191, 191, 192, 192, 192, 192], [1.3264671234316998e6, -689547.4480866259, 121851.83767253702, 1106.4194792955573, 1.1264724564538475e6, -563624.8799894595, 84815.78823219398, 5550.745412136722, 814905.1535154193, -367452.8744356344 … 367452.8744356352, -814905.1535154206, -5550.745412136777, -84815.78823219352, 563624.879989458, -1.126472456453845e6, -1106.4194792956114, -121851.83767253657, 689547.4480866244, -1.3264671234316975e6], 34, 192)], SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}(sparse([1, 2, 3, 4, 5, 1, 2, 3, 4, 5 … 30, 31, 32, 33, 34, 30, 31, 32, 33, 34], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 33, 33, 33, 33, 33, 34, 34, 34, 34, 34], [0.005183531746031743, 0.003844590498236331, 0.0011684303350970016, 0.00012684978505291003, 2.583498677248677e-7, 0.003844590498236331, 0.0072647982804232795, 0.005463468180482069, 0.0015773981757054674, 5.743978725749559e-5 … 5.7439787257495566e-5, 0.0015773981757054678, 0.005463468180482069, 0.007264798280423279, 0.0038445904982363343, 2.583498677248677e-7, 0.00012684978505290995, 0.0011684303350970016, 0.0038445904982363343, 0.005183531746031763], 34, 34), LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}(BandedMatrix(-4 => [7.915e-321, 9.2349344893923e-311, 1.41240070930744e-310, 1.9013092130291e-310, 2.39021636978424e-310, 2.87912492531244e-310, 3.3680317194201e-310, 3.85694063759577e-310, 2.39021750952634e-310, 4.8347567643314e-310 … 1.02127416535986e-309, 1.070164917299606e-309, 1.119055700323446e-309, 1.167946483347286e-309, 1.216837266371126e-309, 1.26572808047906e-309, 1.3146188635029e-309, 1.36350964652674e-309, 1.41240042955058e-309, 1.46129121257442e-309], -3 => [6.9525070095186e-310, 1.62969276746133e-310, 1.35807761705873e-310, 2.60750842793813e-310, 2.33589327753553e-310, 3.5853240884149e-310, 3.31370862717137e-310, 3.80261754534706e-310, 2.44454060177505e-310, 4.7804336720827e-310 … 1.064732608074735e-309, 1.113623391098575e-309, 1.162514174122415e-309, 1.211404957146255e-309, 1.26029577125419e-309, 1.30918655427803e-309, 1.35807733730187e-309, 1.40696812032571e-309, 1.45585890334955e-309, 1.50474968637339e-309], -2 => [0.0, 1.35807730621777e-310, 1.3037550946802e-310, 2.33589296669457e-310, 2.2815708587701e-310, 3.31370862717137e-310, 3.2593865710534e-310, 3.74829445309835e-310, 4.2372015062404e-310, 2.5531867862725e-310 … 1.108191081873704e-309, 1.15708189598164e-309, 1.205972647921384e-309, 1.25486346202932e-309, 1.30375424505316e-309, 1.352645028077e-309, 1.40153581110084e-309, 1.45042659412468e-309, 1.49931737714852e-309, 1.54820816017236e-309], -1 => [2.121995791e-314, 1.4124003984665e-310, 1.2494320024315e-310, 2.3902160589433e-310, 2.22724776652137e-310, 3.3680317194201e-310, 3.2050634788047e-310, 3.69397136084964e-310, 4.1828784139917e-310, 2.49886369402376e-310 … 1.15164958675677e-309, 1.20054033869651e-309, 1.24943115280445e-309, 1.29832193582829e-309, 1.34721271885213e-309, 1.39610350187597e-309, 1.444994284899807e-309, 1.493885067923647e-309, 1.542775850947487e-309, 6.9318993022371e-310] … 1 => [0.05339950200997501, 0.06919570670333844, 0.08129130059963267, 0.07769876040520966, 0.07596526278769833, 0.07525731371417686, 0.0749857577741234, 0.07488400140694504, 0.0748461986255598, 0.07483219899795321 … 0.07482398264448217, 0.074823982639873, 0.07482398263816917, 0.07482398263753932, 0.07482398263730648, 0.07482398263722043, 0.07482398263718862, 0.07033589912319373, 0.06168728222350094, 0.0509054886871495], 2 => [0.01622893206861708, 0.02232810936058315, 0.017125614615854684, 0.015161649485807751, 0.014410209200956373, 0.014124867875957535, 0.014017810251201041, 0.01397795555543467, 0.013963177193647066, 0.013957707100552946 … 0.01395449718093133, 0.013954497176060234, 0.013954497174259575, 0.013954497173593943, 0.013954497173347888, 0.013954497173256928, 0.013954497173223307, 0.013790425533045296, 0.017466024003154612, 0.015569538813896857], 3 => [0.0017618821445366272, 0.0008617480517530313, 0.0006231909324020936, 0.0005416340708845477, 0.0005113557543395947, 0.0004999718067040308, 0.000495714191523132, 0.0004941307575504276, 0.0004935437863948101, 0.0004933265429763531 … 0.0004931990659464415, 0.0004931990654231301, 0.0004931990652296823, 0.0004931990651581722, 0.0004931990651317377, 0.0004931990651219659, 0.0004931990651183537, 0.0004928710169104208, 0.0006553024160425196, 0.001447120422895919], 4 => [3.588354673190687e-6, 1.7284006470886657e-6, 1.2456895686734023e-6, 1.0814856167981285e-6, 1.020620809553661e-6, 9.977491854621264e-7, 9.891966228756468e-7, 9.860160402886302e-7, 9.848370331360776e-7, 9.844006738578623e-7 … 9.84144624301993e-7, 9.841446214584909e-7, 9.841446204073596e-7, 9.841446200187975e-7, 9.841446198751612e-7, 9.841446198220645e-7, 9.841446198024369e-7, 9.841446197951812e-7, 1.3121928263899985e-6, 2.9524338593745232e-6]), 'U', 0), [1.5e-323, 2.0e-323, 4.4e-323, 5.0e-323, 6.0e-323, 1.0e-322, 1.1e-322, 1.7e-322, 2.0e-322, 2.2e-322 … 7.4e-322, 7.7e-322, 7.8e-322, 8.9e-322, 6.9318036259046e-310, 6.93188970734786e-310, 6.9318811704247e-310, 6.93188970734786e-310, 6.93188610382197e-310, 6.93188970734865e-310]), [0.0125, 0.01875, 0.024999999999999998, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997, 0.031249999999999997 … 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.024999999999999998, 0.018750000000000006, 0.01250000000000003], [5.0e-324, 6.931880312107e-310, 5.0e-324, 6.9318803121228e-310, 5.0e-324, 6.9318803121244e-310, 5.0e-324, 6.93188031212597e-310, 5.0e-324, 6.93188031212755e-310 … 5.0e-324, 6.9318803124406e-310, 5.0e-324, 6.93188031244217e-310, 5.0e-324, 6.93188031244533e-310, 5.0e-324, 6.93188031245324e-310, 0.0, 0.0], Dict{Tuple{Int64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}((1, 1) => sparse([1, 2, 3, 4, 5, 1, 2, 3, 4, 5 … 30, 31, 32, 33, 34, 30, 31, 32, 33, 34], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 33, 33, 33, 33, 33, 34, 34, 34, 34, 34], [56.68571428571428, 4.241269841269841, -8.126984126984127, -2.038095238095238, -0.019047619047619046, 4.2412698412698395, 20.26666666666666, 2.116402116402116, -6.38095238095238, -0.9841269841269841 … -0.9841269841269837, -6.380952380952385, 2.1164021164021234, 20.26666666666672, 4.241269841269884, -0.01904761904761907, -2.038095238095237, -8.126984126984123, 4.241269841269883, 56.685714285713445], 34, 34))), SimpleSplines.BandedMass{Float64, SparseArrays.SparseMatrixCSC{Float64, Int64}, LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}}(sparse([1, 2, 3, 4, 5, 1, 2, 3, 4, 5 … 30, 31, 32, 33, 34, 30, 31, 32, 33, 34], [1, 1, 1, 1, 1, 2, 2, 2, 2, 2 … 33, 33, 33, 33, 33, 34, 34, 34, 34, 34], [56.68571428571428, 4.241269841269841, -8.126984126984127, -2.038095238095238, -0.019047619047619046, 4.2412698412698395, 20.26666666666666, 2.116402116402116, -6.38095238095238, -0.9841269841269841 … -0.9841269841269837, -6.380952380952385, 2.1164021164021234, 20.26666666666672, 4.241269841269884, -0.01904761904761907, -2.038095238095237, -8.126984126984123, 4.241269841269883, 56.685714285713445], 34, 34), LinearAlgebra.Cholesky{Float64, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}(BandedMatrix(-4 => [0.0, 0.0, 2.12199579106e-313, 1.976e-321, 2.0e-321, 0.0, 0.0, 2.12199579106e-313, 5.19442209e-315, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], -3 => [0.0, 2.12199579106e-313, 1.5e-321, 1.88e-321, 0.0, 0.0, 2.12199579106e-313, 2.69e-321, 3.5e-323, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], -2 => [2.121995791e-313, 6.3e-322, 1.76e-321, 0.0, 0.0, 2.12199579106e-313, 2.45e-321, 2.4e-321, 5.213080303e-315, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], -1 => [0.0, 1.645e-321, 0.0, 0.0, 2.12199579106e-313, 2.45e-321, 2.28e-321, 0.0, 5.19442122e-315, 5.19442383e-315 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] … 1 => [0.5633250872225498, 0.6099830583227883, 0.37176757297357127, -0.19291010456079447, -0.44644490914259577, -0.5802615652163877, -0.6632195366954241, -0.7209281826003253, -0.764078753790848, -0.7978393728643668 … -0.9678498551489172, -0.9725548685970028, -0.9768908865726382, -0.9808996867647048, -0.9846169684636725, -0.988073419388824, -0.9912955654157596, -0.43568100990373904, 0.20680710861295187, 0.8760349908037105], 2 => [-1.0794253168336285, -1.3944936469468723, -1.5577564963152182, -1.631141537148796, -1.6853169386868467, -1.7260650507341404, -1.756577486700663, -1.7797581390954182, -1.7977758750948098, -1.8121173399554853 … -1.8834574437941405, -1.8856973187830608, -1.887755514527249, -1.8896532767152596, -1.8914086622745436, -1.8930371160534596, -1.894551926708366, -1.8209240261369113, -2.020774159836443, -2.5111944711857626], 3 => [-0.27069963023718346, -0.2200176843312857, -0.20510354995705124, -0.20449444602883993, -0.20753973726541464, -0.21083854143679664, -0.21359999469324972, -0.21578591673647865, -0.21751265310342335, -0.21889628956027177 … -0.22558404955912917, -0.2258223028297131, -0.22604045373627044, -0.2262409437351307, -0.22642583426930996, -0.22659687794244068, -0.22675557427246543, -0.22626385489768283, -0.29824303608756647, -0.618139356745621], 4 => [-0.0025299030863288174, -0.0018953700108259052, -0.001745791322945976, -0.001734506362261369, -0.001758026178911818, -0.0017848974577773635, -0.0018076705118942574, -0.0018257726024655148, -0.0018400947791270554, -0.001851578587731143 … -0.0019049475207505598, -0.0019071178545472741, -0.0019090970372524579, -0.0019109092598020019, -0.001912574795991863, -0.001914110764562973, -0.0019155317200177098, -0.0019168501154547557, -0.0025574355581882793, -0.005757661962390888]), 'U', 0), [1.5e-323, 2.0e-323, 4.4e-323, 5.0e-323, 6.0e-323, 1.0e-322, 1.1e-322, 1.7e-322, 2.0e-322, 2.2e-322 … 7.3e-322, 7.4e-322, 7.7e-322, 7.8e-322, 8.9e-322, 6.93188970734786e-310, 6.9318036259062e-310, 6.93188970734786e-310, 6.9318036259078e-310, 6.93188970734786e-310])), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0])plot(xlabel = "x", ylabel = "ϕ(x)")
plot!(x, p.(x); xlims = domain, label = "Solution")
plot!(x, dsol.(x); xlims = domain, label = "Reference")
The boundary condition is built into the basis rather than imposed on the solution, so it holds exactly:
p(domain[begin]), p(domain[end])(0.0, 0.0)