diff --git a/chapter1/complex_mode.ipynb b/chapter1/complex_mode.ipynb index 6adc9e6c..7958e02d 100644 --- a/chapter1/complex_mode.ipynb +++ b/chapter1/complex_mode.ipynb @@ -11,19 +11,19 @@ "\n", "Many PDEs, such as the [Helmholtz equation](https://docs.fenicsproject.org/dolfinx/v0.4.1/python/demos/demo_helmholtz.html) require complex-valued fields.\n", "\n", - "For simplicity, let us consider a Poisson equation of the form:\n", + "For simplicity, let us consider a Poisson equation of the form: \n", "\n", "$$-\\Delta u = f \\text{ in } \\Omega,$$\n", - "$$ f = -1 - 2j \\text{ in } \\Omega,$$\n", - "$$ u = u*{exact} \\text{ on } \\partial\\Omega,$$\n", - "$$u*{exact}(x, y) = \\frac{1}{2}x^2 + 1j\\cdot y^2,$$\n", + "$$ f = -1 - 2j \\text{ in } \\Omega,$$\n", + "$$ u = u_{exact} \\text{ on } \\partial\\Omega,$$\n", + "$$u_{exact}(x, y) = \\frac{1}{2}x^2 + 1j\\cdot y^2,$$\n", "\n", "As in [Solving the Poisson equation](./fundamentals) we want to express our partial differential equation as a weak formulation.\n", "\n", "We start by defining our discrete function space $V_h$, such that $u_h\\in V_h$ and $u_h = \\sum_{i=1}^N c_i \\phi_i(x, y)$ where $\\phi_i$ are **real valued** global basis functions of our space $V_h$, $c_i \\in \\mathcal{C}$ are the **complex valued** degrees of freedom.\n", "\n", - "Next, we choose a test function $v\\in \\hat V_h$ where $\\hat V_h\\subset V_h$ such that $v\\vert_{\\partial\\Omega}=0$, as done in the first tutorial.\n", - "We now need to define our inner product space. We choose the $L^2$ inner product spaces, which is a _[sesquilinear](https://en.wikipedia.org/wiki/Sesquilinear_form) 2-form_, Meaning that $\\langle u, v\\rangle$ is a map from $V_h\\times V_h\\mapsto K$, and $\\langle u, v \\rangle = \\int_\\Omega u \\cdot \\bar v ~\\mathrm{d} x$. As it is sesquilinear, we have the following properties:\n", + "Next, we choose a test function $v\\in \\hat V_h$ where $\\hat V_h\\subset V_h$ such that $v\\vert_{\\partial\\Omega}=0$, as done in the first tutorial. \n", + "We now need to define our inner product space. We choose the $L^2$ inner product spaces, which is a *[sesquilinear](https://en.wikipedia.org/wiki/Sesquilinear_form) 2-form*, Meaning that $\\langle u, v\\rangle$ is a map from $V_h\\times V_h\\mapsto K$, and $\\langle u, v \\rangle = \\int_\\Omega u \\cdot \\bar v ~\\mathrm{d} x$. As it is sesquilinear, we have the following properties:\n", "\n", "$$\\langle u , v \\rangle = \\overline{\\langle v, u \\rangle},$$\n", "$$\\langle u , u \\rangle \\geq 0.$$\n", @@ -33,8 +33,7 @@ "$$\\int_\\Omega \\nabla u_h \\cdot \\nabla \\overline{v}~ \\mathrm{dx} = \\int_{\\Omega} f \\cdot \\overline{v} ~\\mathrm{d} s \\qquad \\forall v \\in \\hat{V}_h.$$\n", "\n", "## Installation of FEniCSx with complex number support\n", - "\n", - "FEniCSx supports both real and complex numbers, meaning that we can create a function spaces with real valued or complex valued coefficients.\n" + "FEniCSx supports both real and complex numbers, meaning that we can create a function spaces with real valued or complex valued coefficients." ] }, { @@ -71,11 +70,11 @@ "id": "699ab6a9-4427-4c54-95c8-ce5de5d11b97", "metadata": {}, "source": [ - "However, as we would like to solve linear algebra problems on the form $Ax=b$, we need to be able to use matrices and vectors that support real and complex numbers. As [PETSc](https://petsc.org/release/) is one of the most popular interfaces to linear algebra packages, we need to be able to work with their matrix and vector structures.\n", + "However, as we would like to solve linear algebra problems on the form $Ax=b$, we need to be able to use matrices and vectors that support real and complex numbers. As [PETSc](https://petsc.org/release/) is one of the most popular interfaces to linear algebra packages, we need to be able to work with their matrix and vector structures. \n", "\n", "Unfortunately, PETSc only supports one floating type in their matrices, thus we need to install two versions of PETSc, one that supports `float64` and one that supports `complex128`. In the [docker images](https://hub.docker.com/r/dolfinx/dolfinx) for DOLFINx, both versions are installed, and one can switch between them by calling `source dolfinx-real-mode` or `source dolfinx-complex-mode`. For the `dolfinx/lab` images, one can change the Python kernel to be either the real or complex mode, by going to `Kernel->Change Kernel...` and choose `Python3 (ipykernel)` (for real mode) or `Python3 (DOLFINx complex)` (for complex mode).\n", "\n", - "We check that we are using the correct installation of PETSc by inspecting the scalar type.\n" + "We check that we are using the correct installation of PETSc by inspecting the scalar type." ] }, { @@ -105,8 +104,7 @@ "metadata": {}, "source": [ "## Variational problem\n", - "\n", - "We are now ready to define our variational problem\n" + "We are now ready to define our variational problem" ] }, { @@ -133,7 +131,7 @@ "\n", "Secondly, note that we are using `ufl.inner` to describe multiplication of $f$ and $v$, even if they are scalar values. This is because `ufl.inner` takes the conjugate of the second argument, as decribed by the $L^2$ inner product. One could alternatively write this out manually\n", "\n", - "### Inner-products and derivatives\n" + "### Inner-products and derivatives" ] }, { @@ -162,7 +160,7 @@ "id": "9efe0968-bf32-4184-85f7-4e8cc3401cfb", "metadata": {}, "source": [ - "Similarly, if we want to use the function $ufl.derivative$ to take derivatives of functionals, we need to take some special care. As `derivative` inserts a `ufl.TestFunction` to represent the variation, we need to take the conjugate of this to be able to use it to assemble vectors.\n" + "Similarly, if we want to use the function $ufl.derivative$ to take derivatives of functionals, we need to take some special care. As `derivative` inserts a `ufl.TestFunction` to represent the variation, we need to take the conjugate of this to be able to use it to assemble vectors." ] }, { @@ -252,8 +250,7 @@ "metadata": {}, "source": [ "We define our Dirichlet condition and setup and solve the variational problem.\n", - "\n", - "## Solve variational problem\n" + "## Solve variational problem" ] }, { @@ -277,8 +274,7 @@ "metadata": {}, "source": [ "We compute the $L^2$ error and the max error\n", - "\n", - "## Error computation\n" + "## Error computation" ] }, { @@ -298,7 +294,7 @@ "source": [ "x = ufl.SpatialCoordinate(mesh)\n", "u_ex = 0.5 * x[0]**2 + 1j*x[1]**2\n", - "L2_error = dolfinx.fem.form(ufl.dot(uh-u_ex, uh-u_ex) * ufl.dx(metadata={\"quadrature_degree\": 5}))\n", + "L2_error = dolfinx.fem.form(ufl.dot(uh-u_ex, ufl.conj(uh-u_ex)) * ufl.dx(metadata={\"quadrature_degree\": 5}))\n", "local_error = dolfinx.fem.assemble_scalar(L2_error)\n", "global_error = np.sqrt(mesh.comm.allreduce(local_error, op=MPI.SUM))\n", "max_error = mesh.comm.allreduce(np.max(np.abs(u_c.x.array-uh.x.array)))\n", @@ -311,8 +307,7 @@ "metadata": {}, "source": [ "## Plotting\n", - "\n", - "Finally, we plot the real and imaginary solution\n" + "Finally, we plot the real and imaginary solution" ] }, { @@ -340,7 +335,291 @@ "outputs": [ { "data": { - "application/javascript": "(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n var py_version = '3.2.1'.replace('rc', '-rc.').replace('.dev', '-dev.');\n var is_dev = py_version.indexOf(\"+\") !== -1 || py_version.indexOf(\"-\") !== -1;\n var reloading = false;\n var Bokeh = root.Bokeh;\n var bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n\n if (typeof (root._bokeh_timeout) === \"undefined\" || force) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n function run_callbacks() {\n try {\n 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'https://cdn.jsdelivr.net/npm/gridstack@7.2.3/dist/gridstack-all', 'notyf': 'https://cdn.jsdelivr.net/npm/notyf@3/notyf.min'}, 'shim': {'vtk': {'exports': 'vtk'}, 'jspanel': {'exports': 'jsPanel'}, 'gridstack': {'exports': 'GridStack'}}});\n", + " require([\"vtk\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel\"], function(jsPanel) {\n", + "\twindow.jsPanel = jsPanel\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-modal\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-tooltip\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-hint\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-layout\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-contextmenu\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-dock\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"gridstack\"], function(GridStack) {\n", + "\twindow.GridStack = GridStack\n", + "\ton_load()\n", + " })\n", + " require([\"notyf\"], function() {\n", + "\ton_load()\n", + " })\n", + " root._bokeh_is_loading = css_urls.length + 10;\n", + " } else {\n", + " root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n", + " }\n", + "\n", + " var existing_stylesheets = []\n", + " var links = document.getElementsByTagName('link')\n", + " for (var i = 0; i < links.length; i++) {\n", + " var link = links[i]\n", + " if (link.href != null) {\n", + "\texisting_stylesheets.push(link.href)\n", + " }\n", + " }\n", + " for (var i = 0; i < css_urls.length; i++) {\n", + " var url = css_urls[i];\n", + " if (existing_stylesheets.indexOf(url) !== -1) {\n", + "\ton_load()\n", + "\tcontinue;\n", + " }\n", + " const element = document.createElement(\"link\");\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.rel = \"stylesheet\";\n", + " element.type = \"text/css\";\n", + " element.href = url;\n", + " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", + " document.body.appendChild(element);\n", + " } if (((window['vtk'] !== undefined) && (!(window['vtk'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['Notyf'] !== undefined) && (!(window['Notyf'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } var existing_scripts = []\n", + " var scripts = document.getElementsByTagName('script')\n", + " for (var i = 0; i < scripts.length; i++) {\n", + " var script = scripts[i]\n", + " if (script.src != null) {\n", + "\texisting_scripts.push(script.src)\n", + " }\n", + " }\n", + " for (var i = 0; i < js_urls.length; i++) {\n", + " var url = js_urls[i];\n", + " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.src = url;\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " document.head.appendChild(element);\n", + " }\n", + " for (var i = 0; i < js_modules.length; i++) {\n", + " var url = js_modules[i];\n", + " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.src = url;\n", + " element.type = \"module\";\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " document.head.appendChild(element);\n", + " }\n", + " for (const name in js_exports) {\n", + " var url = js_exports[name];\n", + " if (skip.indexOf(url) >= 0 || root[name] != null) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.type = \"module\";\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " element.textContent = `\n", + " import ${name} from \"${url}\"\n", + " window.${name} = ${name}\n", + " window._bokeh_on_load()\n", + " `\n", + " document.head.appendChild(element);\n", + " }\n", + " if (!js_urls.length && !js_modules.length) {\n", + " on_load()\n", + " }\n", + " };\n", + "\n", + " function inject_raw_css(css) {\n", + " const element = document.createElement(\"style\");\n", + " element.appendChild(document.createTextNode(css));\n", + " document.body.appendChild(element);\n", + " }\n", + "\n", + " var js_urls = [\"https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.holoviz.org/panel/1.2.1/dist/panel.min.js\"];\n", + " var js_modules = [];\n", + " var js_exports = {};\n", + " var css_urls = [];\n", + " var inline_js = [ function(Bokeh) {\n", + " Bokeh.set_log_level(\"info\");\n", + " },\n", + "function(Bokeh) {} // ensure no trailing comma for IE\n", + " ];\n", + "\n", + " function run_inline_js() {\n", + " if ((root.Bokeh !== undefined) || (force === true)) {\n", + " for (var i = 0; i < inline_js.length; i++) {\n", + " inline_js[i].call(root, root.Bokeh);\n", + " }\n", + " // Cache old bokeh versions\n", + " if (Bokeh != undefined && !reloading) {\n", + "\tvar NewBokeh = root.Bokeh;\n", + "\tif (Bokeh.versions === undefined) {\n", + "\t Bokeh.versions = new Map();\n", + "\t}\n", + "\tif (NewBokeh.version !== Bokeh.version) {\n", + "\t Bokeh.versions.set(NewBokeh.version, NewBokeh)\n", + "\t}\n", + "\troot.Bokeh = Bokeh;\n", + " }} else if (Date.now() < root._bokeh_timeout) {\n", + " setTimeout(run_inline_js, 100);\n", + " } else if (!root._bokeh_failed_load) {\n", + " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", + " root._bokeh_failed_load = true;\n", + " }\n", + " root._bokeh_is_initializing = false\n", + " }\n", + "\n", + " function load_or_wait() {\n", + " // Implement a backoff loop that tries to ensure we do not load multiple\n", + " // versions of Bokeh and its dependencies at the same time.\n", + " // In recent versions we use the root._bokeh_is_initializing flag\n", + " // to determine whether there is an ongoing attempt to initialize\n", + " // bokeh, however for backward compatibility we also try to ensure\n", + " // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n", + " // before older versions are fully initialized.\n", + " if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n", + " root._bokeh_is_initializing = false;\n", + " root._bokeh_onload_callbacks = undefined;\n", + " console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n", + " load_or_wait();\n", + " } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n", + " setTimeout(load_or_wait, 100);\n", + " } else {\n", + " Bokeh = root.Bokeh;\n", + " bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n", + " root._bokeh_is_initializing = true\n", + " root._bokeh_onload_callbacks = []\n", + " if (!reloading && (!bokeh_loaded || is_dev)) {\n", + "\troot.Bokeh = undefined;\n", + " }\n", + " load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n", + "\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", + "\trun_inline_js();\n", + " });\n", + " }\n", + " }\n", + " // Give older versions of the autoload script a head-start to ensure\n", + " // they initialize before we start loading newer version.\n", + " setTimeout(load_or_wait, 100)\n", + "}(window));" + ], "application/vnd.holoviews_load.v0+json": "" }, "metadata": {}, @@ -348,7 +627,247 @@ }, { "data": { - "application/javascript": "\nif ((window.PyViz === undefined) || (window.PyViz instanceof HTMLElement)) {\n window.PyViz = {comms: {}, comm_status:{}, kernels:{}, receivers: {}, plot_index: []}\n}\n\n\n function JupyterCommManager() {\n }\n\n JupyterCommManager.prototype.register_target = function(plot_id, comm_id, msg_handler) {\n if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n comm_manager.register_target(comm_id, function(comm) {\n comm.on_msg(msg_handler);\n });\n } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n 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{\n", + " toinsert[nchildren-1].children[1].textContent = output.data[JS_MIME_TYPE];\n", + " }\n", + " output_area._hv_plot_id = id;\n", + " if ((window.Bokeh !== undefined) && (id in Bokeh.index)) {\n", + " window.PyViz.plot_index[id] = Bokeh.index[id];\n", + " } else {\n", + " window.PyViz.plot_index[id] = null;\n", + " }\n", + " } else if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n", + " var bk_div = document.createElement(\"div\");\n", + " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", + " var script_attrs = bk_div.children[0].attributes;\n", + " for (var i = 0; i < script_attrs.length; i++) {\n", + " toinsert[toinsert.length - 1].childNodes[1].setAttribute(script_attrs[i].name, script_attrs[i].value);\n", + " }\n", + " // store reference to server id on output_area\n", + " output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n", + " }\n", + "}\n", + "\n", + "/**\n", + " * Handle when an output is cleared or removed\n", + " */\n", + "function handle_clear_output(event, handle) {\n", + " var id = handle.cell.output_area._hv_plot_id;\n", + " var server_id = handle.cell.output_area._bokeh_server_id;\n", + " if (((id === undefined) || !(id in PyViz.plot_index)) && (server_id !== undefined)) { return; }\n", + " var comm = window.PyViz.comm_manager.get_client_comm(\"hv-extension-comm\", \"hv-extension-comm\", function () {});\n", + " if (server_id !== null) {\n", + " comm.send({event_type: 'server_delete', 'id': server_id});\n", + " return;\n", + " } else if (comm !== null) {\n", + " comm.send({event_type: 'delete', 'id': id});\n", + " }\n", + " delete PyViz.plot_index[id];\n", + " if ((window.Bokeh !== undefined) & (id in window.Bokeh.index)) {\n", + " var doc = window.Bokeh.index[id].model.document\n", + " doc.clear();\n", + " const i = window.Bokeh.documents.indexOf(doc);\n", + " if (i > -1) {\n", + " window.Bokeh.documents.splice(i, 1);\n", + " }\n", + " }\n", + "}\n", + "\n", + "/**\n", + " * Handle kernel restart event\n", + " */\n", + "function handle_kernel_cleanup(event, handle) {\n", + " delete PyViz.comms[\"hv-extension-comm\"];\n", + " window.PyViz.plot_index = {}\n", + "}\n", + "\n", + "/**\n", + " * Handle update_display_data messages\n", + " */\n", + "function handle_update_output(event, handle) {\n", + " handle_clear_output(event, {cell: {output_area: handle.output_area}})\n", + " handle_add_output(event, handle)\n", + "}\n", + "\n", + "function register_renderer(events, OutputArea) {\n", + " function append_mime(data, metadata, element) {\n", + " // create a DOM node to render to\n", + " var toinsert = this.create_output_subarea(\n", + " metadata,\n", + " CLASS_NAME,\n", + " EXEC_MIME_TYPE\n", + " );\n", + " this.keyboard_manager.register_events(toinsert);\n", + " // Render to node\n", + " var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n", + " render(props, toinsert[0]);\n", + " element.append(toinsert);\n", + " return toinsert\n", + " }\n", + "\n", + " events.on('output_added.OutputArea', handle_add_output);\n", + " events.on('output_updated.OutputArea', handle_update_output);\n", + " events.on('clear_output.CodeCell', handle_clear_output);\n", + " events.on('delete.Cell', handle_clear_output);\n", + " events.on('kernel_ready.Kernel', handle_kernel_cleanup);\n", + "\n", + " OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n", + " safe: true,\n", + " index: 0\n", + " });\n", + "}\n", + "\n", + "if (window.Jupyter !== undefined) {\n", + " try {\n", + " var events = require('base/js/events');\n", + " var OutputArea = require('notebook/js/outputarea').OutputArea;\n", + " if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n", + " register_renderer(events, OutputArea);\n", + " }\n", + " } catch(err) {\n", + " }\n", + "}\n" + ], "application/vnd.holoviews_load.v0+json": "" }, "metadata": {}, @@ -672,9 +1191,9 @@ "formats": "ipynb,py:light" }, "kernelspec": { - "display_name": "Python 3 (DOLFINx complex)", + "display_name": "Python 3 (ipykernel)", "language": "python", - "name": "python3-complex" + "name": "python3" }, "language_info": { "codemirror_mode": { @@ -686,7 +1205,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.10.6" + "version": "3.10.12" } }, "nbformat": 4, diff --git a/chapter1/complex_mode.py b/chapter1/complex_mode.py index 77bb41a4..5ca55a2c 100644 --- a/chapter1/complex_mode.py +++ b/chapter1/complex_mode.py @@ -108,7 +108,7 @@ x = ufl.SpatialCoordinate(mesh) u_ex = 0.5 * x[0]**2 + 1j*x[1]**2 -L2_error = dolfinx.fem.form(ufl.dot(uh-u_ex, uh-u_ex) * ufl.dx(metadata={"quadrature_degree": 5})) +L2_error = dolfinx.fem.form(ufl.dot(uh-u_ex, ufl.conj(uh-u_ex)) * ufl.dx(metadata={"quadrature_degree": 5})) local_error = dolfinx.fem.assemble_scalar(L2_error) global_error = np.sqrt(mesh.comm.allreduce(local_error, op=MPI.SUM)) max_error = mesh.comm.allreduce(np.max(np.abs(u_c.x.array-uh.x.array)))