![](\"https://raw.githack.com/spectralDNS/spectralutilities/master/figures/poisson1D_errornorm.png\")
Figure 1: Convergence of 1D Poisson solvers for both Legendre and Chebyshev modified basis function.
\n", "" ] }, { "cell_type": "markdown", "id": "5c9e928b", "metadata": { "editable": true }, "source": [ "## Poisson's equation\n", "\n", "Poisson's equation is given as" ] }, { "cell_type": "markdown", "id": "88a950ff", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\nabla^2 u(x) = f(x) \\quad \\text{for }\\, x \\in (-1, 1), \\label{eq:poisson} \\tag{1}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "04929b21", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \n", "u(-1)=a, u(1)=b, \\notag\n", "\\label{_auto1} \\tag{2}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "6dff101a", "metadata": { "editable": true }, "source": [ "where $u(x)$ is the solution, $f(x)$ is a function and $a, b$ are two possibly\n", "non-zero constants.\n", "\n", "To solve Eq. ([1](#eq:poisson)) with the Galerkin method we need smooth continuously\n", "differentiable basis functions, $v_k$, that satisfy the given boundary conditions.\n", "And then we look for solutions like" ] }, { "cell_type": "markdown", "id": "40cbe4bb", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "u(x) = \\sum_{k=0}^{N-1} \\hat{u}_k v_k(x), \\label{eq:u} \\tag{3}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "75b794d4", "metadata": { "editable": true }, "source": [ "where $N$ is the size of the discretized problem,\n", "$\\{\\hat{u}_k\\}_{k=0}^{N-1}$ are the unknown expansion\n", "coefficients, and the function space is $\\text{span}\\{v_k\\}_{k=0}^{N-1}$.\n", "\n", "The basis functions of the function space can, for example, be constructed from\n", "[Chebyshev](https://en.wikipedia.org/wiki/Chebyshev_polynomials), $T_k(x)$, or\n", "[Legendre](https://en.wikipedia.org/wiki/Legendre_polynomials), $L_k(x)$, polynomials\n", "and we use the common notation $\\phi_k(x)$ to represent either one of them. It turns out that\n", "it is easiest to use basis functions with homogeneous Dirichlet boundary conditions" ] }, { "cell_type": "markdown", "id": "daad7929", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "v_k(x) = \\phi_k(x) - \\phi_{k+2}(x),\n", "\\label{_auto2} \\tag{4}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "49939654", "metadata": { "editable": true }, "source": [ "for $k=0, 1, \\ldots N-3$. This gives the function space\n", "$V^N_0 = \\text{span}\\{v_k(x)\\}_{k=0}^{N-3}$.\n", "We can then add two more linear basis functions (that belong to the kernel of Poisson's equation)" ] }, { "cell_type": "markdown", "id": "c2eb52a2", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "v_{N-2} = \\frac{1}{2}(\\phi_0 - \\phi_1), \n", "\\label{_auto3} \\tag{5}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "19b3c0c6", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \n", "v_{N-1} = \\frac{1}{2}(\\phi_0 + \\phi_1).\n", "\\label{_auto4} \\tag{6}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "f9753564", "metadata": { "editable": true }, "source": [ "which gives the inhomogeneous space $V^N = \\text{span}\\{v_k\\}_{k=0}^{N-1}$.\n", "With the two linear basis functions it is easy to see that the last two degrees\n", "of freedom, $\\hat{u}_{N-2}$ and $\\hat{u}_{N-1}$, now are given as" ] }, { "cell_type": "markdown", "id": "7ee320eb", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "u(-1) = \\sum_{k=0}^{N-1} \\hat{u}_k v_k(-1) = \\hat{u}_{N-2} = a,\n", "\\label{eq:dirichleta} \\tag{7} \n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "5bfbba11", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \n", "u(+1) = \\sum_{k=0}^{N-1} \\hat{u}_k v_k(+1) = \\hat{u}_{N-1} = b,\n", "\\label{eq:dirichletb} \\tag{8}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "a377dc2d", "metadata": { "editable": true }, "source": [ "and, as such, we only have to solve for $\\{\\hat{u}_k\\}_{k=0}^{N-3}$, just like\n", "for a problem with homogeneous boundary conditions (for homogeneous boundary condition\n", "we simply have $\\hat{u}_{N-2} = \\hat{u}_{N-1} = 0$).\n", "We now formulate a variational problem using the Galerkin method: Find $u \\in V^N$ such that" ] }, { "cell_type": "markdown", "id": "bc16986e", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\int_{-1}^1 \\nabla^2 u \\, v \\, w\\, dx = \\int_{-1}^1 f \\, v\\, w\\, dx \\quad \\forall v \\, \\in \\, V^N_0. \\label{eq:varform} \\tag{9}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "c56593b5", "metadata": { "editable": true }, "source": [ "Note that since we only have $N-3$ unknowns we are only using the homogeneous test\n", "functions from $V^N_0$.\n", "\n", "The weighted integrals, weighted by $w(x)$, are called inner products, and a\n", "common notation is" ] }, { "cell_type": "markdown", "id": "e7f282db", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "\\int_{-1}^1 u \\, v \\, w\\, dx = \\left( u, v\\right)_w.\n", "\\label{_auto5} \\tag{10}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "2aae170f", "metadata": { "editable": true }, "source": [ "The integral can either be computed exactly, or with quadrature. The advantage\n", "of the latter is that it is generally faster, and that non-linear terms may be\n", "computed just as quickly as linear. For a linear problem, it does not make much\n", "of a difference, if any at all. Approximating the integral with quadrature, we\n", "obtain" ] }, { "cell_type": "markdown", "id": "0597485b", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", "\\int_{-1}^1 u \\, v \\, w\\, dx &\\approx \\left( u, v \\right)_w^N, \\\\ \n", "&\\approx \\sum_{j=0}^{N-1} u(x_j) v(x_j) w_j,\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "1820492f", "metadata": { "editable": true }, "source": [ "where $\\{w_j\\}_{j=0}^{N-1}$ are quadrature weights.\n", "The quadrature points $\\{x_j\\}_{j=0}^{N-1}$\n", "are specific to the chosen basis, and even within basis there are two different\n", "choices based on which quadrature rule is selected, either Gauss or Gauss-Lobatto.\n", "\n", "Inserting for test and trialfunctions, we get the following bilinear form and\n", "matrix $A=(a_{jk})\\in\\mathbb{R}^{N-2\\times N-2}$ for the Laplacian" ] }, { "cell_type": "markdown", "id": "0f03aa15", "metadata": { "editable": true }, "source": [ "$$\n", "\\begin{align*}\n", "\\left( \\nabla^2u, v \\right)_w^N &= \\left( \\nabla^2\\sum_{k=0}^{N-3}\\hat{u}_k v_{k}, v_j \\right)_w^N, \\quad j=0,1,\\ldots, N-3\\\\ \n", " &= \\sum_{k=0}^{N-3}\\left(\\nabla^2 v_{k}, v_j \\right)_w^N \\hat{u}_k, \\\\ \n", " &= \\sum_{k=0}^{N-3}a_{jk} \\hat{u}_k.\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "id": "08f3c800", "metadata": { "editable": true }, "source": [ "Note that the sum runs over $k=0, 1, \\ldots, N-3$ since\n", "the second derivatives of $v_{N-2}$ and $v_{N-1}$ are zero.\n", "The right hand side linear form and vector is computed as $\\tilde{f}_j = (f,\n", "v_j)_w^N$, for $j=0,1,\\ldots, N-3$, where a tilde is used because this is not\n", "a complete transform of the function $f$, but only an inner product.\n", "\n", "By defining the column vectors $\\boldsymbol{\\hat{u}}=(\\hat{u}_0, \\hat{u}_1, \\ldots, \\hat{u}_{N-3})^T$\n", "and $\\boldsymbol{\\tilde{f}}=(\\tilde{f}_0, \\tilde{f}_1, \\ldots, \\tilde{f}_{N-3})^T$\n", "we get the linear system of equations" ] }, { "cell_type": "markdown", "id": "65e0bcc0", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "A \\hat{\\boldsymbol{u}} = \\tilde{\\boldsymbol{f}}.\n", "\\label{_auto6} \\tag{11}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "998ba60a", "metadata": { "editable": true }, "source": [ "Now, when the expansion coefficients $\\boldsymbol{\\hat{u}}$ are found by\n", "solving this linear system, they may be\n", "transformed to real space $u(x)$ using ([3](#eq:u)), and here the contributions\n", "from $\\hat{u}_{N-2}$ and $\\hat{u}_{N-1}$ must be accounted for. Note that the matrix\n", "$A$ (different for Legendre or Chebyshev) has a very special structure that\n", "allows for a solution to be found very efficiently in order of $\\mathcal{O}(N)$\n", "operations, see [[shen1]](#shen1) and [[shen95]](#shen95). These solvers are implemented in\n", "shenfun for both bases." ] }, { "cell_type": "markdown", "id": "81e844aa", "metadata": { "editable": true }, "source": [ "## Method of manufactured solutions\n", "\n", "In this demo we will use the method of manufactured\n", "solutions to demonstrate spectral accuracy of the `shenfun` Dirichlet bases. To\n", "this end we choose an analytical function that satisfies the given boundary\n", "conditions:" ] }, { "cell_type": "markdown", "id": "a7b27de2", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", "u_e(x) = \\sin(k\\pi x)(1-x^2) + a(1-x)/2 + b(1+x)/2, \\label{eq:u_e} \\tag{12}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "14a5e799", "metadata": { "editable": true }, "source": [ "where $k$ is an integer and $a$ and $b$ are constants. Now, feeding $u_e$ through\n", "the Laplace operator, we see that the last two linear terms disappear, whereas the\n", "first term results in" ] }, { "cell_type": "markdown", "id": "d45e3287", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation}\n", " \\nabla^2 u_e(x) = \\frac{d^2 u_e}{dx^2}, \n", "\\label{_auto7} \\tag{13}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "d0f28dd7", "metadata": { "editable": true }, "source": [ "\n", "\n", "\n", "$$\n", "\\begin{equation} \n", " = -4k \\pi x \\cos(k\\pi x) - 2\\sin(k\\pi x) - k^2 \\pi^2 (1 -\n", "x^2) \\sin(k \\pi x). \\label{eq:solution} \\tag{14}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", "id": "91f8eb49", "metadata": { "editable": true }, "source": [ "Now, setting $f_e(x) = \\nabla^2 u_e(x)$ and solving for $\\nabla^2 u(x) = f_e(x)$,\n", "we can compare the numerical solution $u(x)$ with the analytical solution $u_e(x)$\n", "and compute error norms." ] }, { "cell_type": "markdown", "id": "fc08f897", "metadata": { "editable": true }, "source": [ "## Implementation" ] }, { "cell_type": "markdown", "id": "dc126f5a", "metadata": { "editable": true }, "source": [ "### Preamble\n", "\n", "We will solve Poisson's equation using the [shenfun](https://github.com/spectralDNS/shenfun) Python module. The first thing needed\n", "is then to import some of this module's functionality\n", "plus some other helper modules, like [Numpy](https://numpy.org) and [Sympy](https://sympy.org):" ] }, { "cell_type": "code", "execution_count": 1, "id": "659b5c0d", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:42.681553Z", "iopub.status.busy": "2026-06-03T08:49:42.680941Z", "iopub.status.idle": "2026-06-03T08:49:43.086984Z", "shell.execute_reply": "2026-06-03T08:49:43.086733Z" }, "tags": [ "thebe-init" ] }, "outputs": [], "source": [ "from shenfun import inner, div, grad, TestFunction, TrialFunction, Function, \\\n", " project, Dx, Array, FunctionSpace, dx\n", "import numpy as np\n", "from sympy import symbols, cos, sin, exp, lambdify" ] }, { "cell_type": "markdown", "id": "cfc9c744", "metadata": { "editable": true }, "source": [ "We use `Sympy` for the manufactured solution and `Numpy` for testing." ] }, { "cell_type": "markdown", "id": "40d4aa04", "metadata": { "editable": true }, "source": [ "### Manufactured solution\n", "\n", "The exact solution $u_e(x)$ and the right hand side $f_e(x)$ are created using\n", "`Sympy` as follows" ] }, { "cell_type": "code", "execution_count": 2, "id": "38c3edf0", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.088919Z", "iopub.status.busy": "2026-06-03T08:49:43.088663Z", "iopub.status.idle": "2026-06-03T08:49:43.100902Z", "shell.execute_reply": "2026-06-03T08:49:43.100680Z" }, "tags": [ "thebe-init" ] }, "outputs": [], "source": [ "a = -1\n", "b = 1\n", "k = 4\n", "x = symbols(\"x\")\n", "ue = sin(k*np.pi*x)*(1-x**2) + a*(1 - x)/2. + b*(1 + x)/2.\n", "fe = ue.diff(x, 2)" ] }, { "cell_type": "markdown", "id": "a66d4067", "metadata": { "editable": true }, "source": [ "These solutions are now valid for a continuous domain. The next step is thus to\n", "discretize, using a discrete mesh $\\{x_j\\}_{j=0}^{N-1}$ and a finite number of\n", "basis functions.\n", "\n", "Note that it is not mandatory to use `Sympy` for the manufactured solution. Since the\n", "solution is known ([14](#eq:solution)), we could just as well simply use `Numpy`\n", "to compute $f_e$ at $\\{x_j\\}_{j=0}^{N-1}$. However, with `Sympy` it is much\n", "easier to experiment and quickly change the solution." ] }, { "cell_type": "markdown", "id": "e2cfe44f", "metadata": { "editable": true }, "source": [ "### Discretization\n", "\n", "We create a basis with a given number of basis functions, and extract the computational\n", "mesh from the basis itself" ] }, { "cell_type": "code", "execution_count": 3, "id": "9328ec31", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.102247Z", "iopub.status.busy": "2026-06-03T08:49:43.102161Z", "iopub.status.idle": "2026-06-03T08:49:43.113237Z", "shell.execute_reply": "2026-06-03T08:49:43.113007Z" }, "tags": [ "thebe-init" ] }, "outputs": [], "source": [ "N = 32\n", "SD = FunctionSpace(N, 'Chebyshev', bc=(a, b))\n", "#SD = FunctionSpace(N, 'Legendre', bc=(a, b))" ] }, { "cell_type": "markdown", "id": "db6ef1ac", "metadata": { "editable": true }, "source": [ "Note that we can either choose a Legendre or a Chebyshev basis." ] }, { "cell_type": "markdown", "id": "37f9d1b5", "metadata": { "editable": true }, "source": [ "### Variational formulation\n", "\n", "The variational problem ([9](#eq:varform)) can be assembled using `shenfun`'s\n", "[TrialFunction](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.TrialFunction), [TestFunction](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.TestFunction) and [inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner) functions." ] }, { "cell_type": "code", "execution_count": 4, "id": "d3f0a739", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.114498Z", "iopub.status.busy": "2026-06-03T08:49:43.114425Z", "iopub.status.idle": "2026-06-03T08:49:43.123259Z", "shell.execute_reply": "2026-06-03T08:49:43.123028Z" }, "tags": [ "thebe-init" ] }, "outputs": [], "source": [ "u = TrialFunction(SD)\n", "v = TestFunction(SD)\n", "# Assemble left hand side matrix\n", "A = inner(v, div(grad(u)))\n", "# Assemble right hand side\n", "fj = Array(SD, buffer=fe)\n", "f_hat = Function(SD)\n", "f_hat = inner(v, fj, output_array=f_hat)" ] }, { "cell_type": "markdown", "id": "5ecdef9b", "metadata": { "editable": true }, "source": [ "Note that the `sympy` function `fe` can be used to initialize the [Array](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.Array)\n", "`fj`. We wrap this Numpy array in an [Array](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.arguments.Array) class\n", "(`fj = Array(SD, buffer=fe)`), because an Array\n", "is required as input to the [inner()](https://shenfun.readthedocs.io/en/latest/shenfun.forms.html#shenfun.forms.inner.inner) function." ] }, { "cell_type": "markdown", "id": "ee8d601e", "metadata": { "editable": true }, "source": [ "### Solve linear equations\n", "\n", "Finally, solve linear equation system and transform solution from spectral\n", "$\\boldsymbol{\\hat{u}}$ vector to the real space $\\{u(x_j)\\}_{j=0}^{N-1}$\n", "and then check how the solution corresponds with the exact solution $u_e$.\n", "To this end we compute the $L_2$-errornorm using the `shenfun` function\n", "[dx()](https://shenfun.readthedocs.io/en/latest/shenfun.utilities.html#shenfun.utilities.dx)" ] }, { "cell_type": "code", "execution_count": 5, "id": "3efea157", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.124585Z", "iopub.status.busy": "2026-06-03T08:49:43.124506Z", "iopub.status.idle": "2026-06-03T08:49:43.137796Z", "shell.execute_reply": "2026-06-03T08:49:43.137606Z" }, "tags": [ "thebe-init" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Error=2.3565371191241489e-10\n" ] } ], "source": [ "u_hat = A.solve(f_hat)\n", "uj = SD.backward(u_hat)\n", "ua = Array(SD, buffer=ue)\n", "print(\"Error=%2.16e\" %(np.sqrt(dx((uj-ua)**2))))" ] }, { "cell_type": "markdown", "id": "3c8908a6", "metadata": { "editable": true }, "source": [ "### Convergence test\n", "\n", "To do a convergence test we will now create a function `main`, that takes the\n", "number of quadrature points as parameter, and prints out\n", "the error." ] }, { "cell_type": "code", "execution_count": 6, "id": "c69c9d1e", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.139045Z", "iopub.status.busy": "2026-06-03T08:49:43.138971Z", "iopub.status.idle": "2026-06-03T08:49:43.141093Z", "shell.execute_reply": "2026-06-03T08:49:43.140875Z" }, "tags": [ "thebe-init" ] }, "outputs": [], "source": [ "def main(N, family='Chebyshev'):\n", " SD = FunctionSpace(N, family=family, bc=(a, b))\n", " u = TrialFunction(SD)\n", " v = TestFunction(SD)\n", "\n", " # Get f on quad points\n", " fj = Array(SD, buffer=fe)\n", "\n", " # Compute right hand side of Poisson's equation\n", " f_hat = Function(SD)\n", " f_hat = inner(v, fj, output_array=f_hat)\n", "\n", " # Get left hand side of Poisson's equation\n", " A = inner(v, div(grad(u)))\n", "\n", " f_hat = A.solve(f_hat)\n", " uj = SD.backward(f_hat)\n", "\n", " # Compare with analytical solution\n", " ua = Array(SD, buffer=ue)\n", " l2_error = np.linalg.norm(uj-ua)\n", " return l2_error" ] }, { "cell_type": "markdown", "id": "8199b9c0", "metadata": { "editable": true }, "source": [ "For example, we find the error of a Chebyshev discretization\n", "using 12 quadrature points as" ] }, { "cell_type": "code", "execution_count": 7, "id": "e2444fa7", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.142236Z", "iopub.status.busy": "2026-06-03T08:49:43.142166Z", "iopub.status.idle": "2026-06-03T08:49:43.149893Z", "shell.execute_reply": "2026-06-03T08:49:43.149669Z" }, "tags": [ "thebe-init" ] }, "outputs": [ { "data": { "text/plain": [ "1.1654916318123927" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "main(12, 'Chebyshev')" ] }, { "cell_type": "markdown", "id": "41161308", "metadata": { "editable": true }, "source": [ "To get the convergence we call `main` for a list\n", "of $N=[12, 16, \\ldots, 48]$, and collect the errornorms in\n", "arrays to be plotted. The error can be plotted using\n", "[matplotlib](https://matplotlib.org), and the generated\n", "figure is also shown in this demos summary." ] }, { "cell_type": "code", "execution_count": 8, "id": "be4c522a", "metadata": { "editable": true, "execution": { "iopub.execute_input": "2026-06-03T08:49:43.151092Z", "iopub.status.busy": "2026-06-03T08:49:43.151020Z", "iopub.status.idle": "2026-06-03T08:49:43.870264Z", "shell.execute_reply": "2026-06-03T08:49:43.870018Z" }, "tags": [ "thebe-init" ] }, "outputs": [ { "data": { "image/png": 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", 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