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Offset 1, 12 lines modified		Offset 1, 12 lines modified
1	{	1	{
2	····"cells":·[	2	····"cells":·[
3	········{	3	········{
4	············"cell_type":·"markdown",	4	············"cell_type":·"markdown",
5	············"id":·"~~795a0907~~",	5	············"id":·"6aed6bf9",
6	············"metadata":·{},	6	············"metadata":·{},
7	············"source":·[	7	············"source":·[
8	················"#·Matrix-free·conjugate·gradient·solver·for·the·Poisson·equation\n",	8	················"#·Matrix-free·conjugate·gradient·solver·for·the·Poisson·equation\n",
9	················"\n",	9	················"\n",
10	················"This·demo·illustrates·how·to·solve·the·Poisson·equation·using·a\n",	10	················"This·demo·illustrates·how·to·solve·the·Poisson·equation·using·a\n",
11	················"matrix-free·conjugate·gradient·(CG)·solver.·In·particular,·it\n",	11	················"matrix-free·conjugate·gradient·(CG)·solver.·In·particular,·it\n",
12	················"illustrates·how·to\n",	12	················"illustrates·how·to\n",
Offset 68, 84 lines modified		Offset 68, 84 lines modified
68	················"\n",	68	················"\n",
69	················"The·modules·that·will·be·used·are·imported:"	69	················"The·modules·that·will·be·used·are·imported:"
70	············]	70	············]
71	········},	71	········},
72	········{	72	········{
73	············"cell_type":·"code",	73	············"cell_type":·"code",
74	············"execution_count":·null,	74	············"execution_count":·null,
75	············"id":·"~~d5c0af21~~",	75	············"id":·"09b20566",
76	············"metadata":·{},	76	············"metadata":·{},
77	············"outputs":·[],	77	············"outputs":·[],
78	············"source":·[	78	············"source":·[
79	················"from·mpi4py·import·MPI"	79	················"from·mpi4py·import·MPI"
80	············]	80	············]
81	········},	81	········},
82	········{	82	········{
83	············"cell_type":·"code",	83	············"cell_type":·"code",
84	············"execution_count":·null,	84	············"execution_count":·null,
85	············"id":·"~~543f01aa~~",	85	············"id":·"3e0b53cf",
86	············"metadata":·{},	86	············"metadata":·{},
87	············"outputs":·[],	87	············"outputs":·[],
88	············"source":·[	88	············"source":·[
89	················"import·numpy·as·np"	89	················"import·numpy·as·np"
90	············]	90	············]
91	········},	91	········},
92	········{	92	········{
93	············"cell_type":·"code",	93	············"cell_type":·"code",
94	············"execution_count":·null,	94	············"execution_count":·null,
95	············"id":·"~~7547dc51~~",	95	············"id":·"349e8e38",
96	············"metadata":·{},	96	············"metadata":·{},
97	············"outputs":·[],	97	············"outputs":·[],
98	············"source":·[	98	············"source":·[
99	················"import·dolfinx\n",	99	················"import·dolfinx\n",
100	················"import·ufl\n",	100	················"import·ufl\n",
101	················"from·dolfinx·import·fem,·la\n",	101	················"from·dolfinx·import·fem,·la\n",
102	················"from·ufl·import·action,·dx,·grad,·inner"	102	················"from·ufl·import·action,·dx,·grad,·inner"
103	············]	103	············]
104	········},	104	········},
105	········{	105	········{
106	············"cell_type":·"markdown",	106	············"cell_type":·"markdown",
107	············"id":·"~~8fbaab00~~",	107	············"id":·"5ddce97e",
108	············"metadata":·{},	108	············"metadata":·{},
109	············"source":·[	109	············"source":·[
110	················"We·begin·by·using·{py:func}`create_rectangle\n",	110	················"We·begin·by·using·{py:func}`create_rectangle\n",
111	················"<dolfinx.mesh.create_rectangle>`·to·create·a·rectangular\n",	111	················"<dolfinx.mesh.create_rectangle>`·to·create·a·rectangular\n",
112	················"{py:class}`Mesh·<dolfinx.mesh.Mesh>`·of·the·domain,·and·creating·a\n",	112	················"{py:class}`Mesh·<dolfinx.mesh.Mesh>`·of·the·domain,·and·creating·a\n",
113	················"finite·element·{py:class}`FunctionSpace·<dolfinx.fem.FunctionSpace>`\n",	113	················"finite·element·{py:class}`FunctionSpace·<dolfinx.fem.FunctionSpace>`\n",
114	················"on·the·mesh."	114	················"on·the·mesh."
115	············]	115	············]
116	········},	116	········},
117	········{	117	········{
118	············"cell_type":·"code",	118	············"cell_type":·"code",
119	············"execution_count":·null,	119	············"execution_count":·null,
120	············"id":·"~~ea86aece~~",	120	············"id":·"2238c328",
121	············"metadata":·{},	121	············"metadata":·{},
122	············"outputs":·[],	122	············"outputs":·[],
123	············"source":·[	123	············"source":·[
124	················"dtype·=·dolfinx.default_scalar_type\n",	124	················"dtype·=·dolfinx.default_scalar_type\n",
125	················"real_type·=·np.real(dtype(0.0)).dtype\n",	125	················"real_type·=·np.real(dtype(0.0)).dtype\n",
126	················"comm·=·MPI.COMM_WORLD\n",	126	················"comm·=·MPI.COMM_WORLD\n",
127	················"mesh·=·dolfinx.mesh.create_rectangle(comm,·[[0.0,·0.0],·[1.0,·1.0]],·[10,·10],·dtype=real_type)"	127	················"mesh·=·dolfinx.mesh.create_rectangle(comm,·[[0.0,·0.0],·[1.0,·1.0]],·[10,·10],·dtype=real_type)"
128	············]	128	············]
129	········},	129	········},
130	········{	130	········{
131	············"cell_type":·"code",	131	············"cell_type":·"code",
132	············"execution_count":·null,	132	············"execution_count":·null,
133	············"id":·"~~e9d92459~~",	133	············"id":·"d06dddfe",
134	············"metadata":·{},	134	············"metadata":·{},
135	············"outputs":·[],	135	············"outputs":·[],
136	············"source":·[	136	············"source":·[
137	················"#·Create·function·space\n",	137	················"#·Create·function·space\n",
138	················"degree·=·2\n",	138	················"degree·=·2\n",
139	················"V·=·fem.functionspace(mesh,·(\"Lagrange\",·degree))"	139	················"V·=·fem.functionspace(mesh,·(\"Lagrange\",·degree))"
140	············]	140	············]
141	········},	141	········},
142	········{	142	········{
143	············"cell_type":·"markdown",	143	············"cell_type":·"markdown",
144	············"id":·"~~c4866470~~",	144	············"id":·"a0c7fbce",
145	············"metadata":·{},	145	············"metadata":·{},
146	············"source":·[	146	············"source":·[
147	················"The·second·argument·to·{py:class}`functionspace\n",	147	················"The·second·argument·to·{py:class}`functionspace\n",
148	················"<dolfinx.fem.functionspace>`·is·a·tuple·consisting·of·`(family,\n",	148	················"<dolfinx.fem.functionspace>`·is·a·tuple·consisting·of·`(family,\n",
149	················"degree)`,·where·`family`·is·the·finite·element·family,·and·`degree`\n",	149	················"degree)`,·where·`family`·is·the·finite·element·family,·and·`degree`\n",
150	················"specifies·the·polynomial·degree.·In·this·case·`V`·consists·of\n",	150	················"specifies·the·polynomial·degree.·In·this·case·`V`·consists·of\n",
151	················"third-order,·continuous·Lagrange·finite·element·functions.\n",	151	················"third-order,·continuous·Lagrange·finite·element·functions.\n",
Offset 156, 94 lines modified		Offset 156, 94 lines modified
156	················"and·then·retrieving·all·facets·on·the·boundary·using\n",	156	················"and·then·retrieving·all·facets·on·the·boundary·using\n",
157	················"{py:func}`exterior_facet_indices·<dolfinx.mesh.exterior_facet_indices>`."	157	················"{py:func}`exterior_facet_indices·<dolfinx.mesh.exterior_facet_indices>`."
158	············]	158	············]
159	········},	159	········},
160	········{	160	········{
161	············"cell_type":·"code",	161	············"cell_type":·"code",
162	············"execution_count":·null,	162	············"execution_count":·null,
163	············"id":·"e2~~ec26~~91",	163	············"id":·"b27ff39a",
164	············"metadata":·{},	164	············"metadata":·{},
165	············"outputs":·[],	165	············"outputs":·[],
166	············"source":·[	166	············"source":·[
167	················"tdim·=·mesh.topology.dim\n",	167	················"tdim·=·mesh.topology.dim\n",
168	················"mesh.topology.create_connectivity(tdim·-·1,·tdim)\n",	168	················"mesh.topology.create_connectivity(tdim·-·1,·tdim)\n",
169	················"facets·=·dolfinx.mesh.exterior_facet_indices(mesh.topology)"	169	················"facets·=·dolfinx.mesh.exterior_facet_indices(mesh.topology)"
170	············]	170	············]
171	········},	171	········},
172	········{	172	········{
173	············"cell_type":·"markdown",	173	············"cell_type":·"markdown",
174	············"id":·"e~~a6b0~~e3a",	174	············"id":·"e71e6e47",
175	············"metadata":·{},	175	············"metadata":·{},
176	············"source":·[	176	············"source":·[
177	················"We·now·find·the·degrees·of·freedom·that·are·associated·with·the·boundary\n",	177	················"We·now·find·the·degrees·of·freedom·that·are·associated·with·the·boundary\n",
178	················"facets·using\n",	178	················"facets·using\n",
179	················"{py:func}`locate_dofs_topological·<dolfinx.fem.locate_dofs_topological>`"	179	················"{py:func}`locate_dofs_topological·<dolfinx.fem.locate_dofs_topological>`"
180	············]	180	············]
181	········},	181	········},
182	········{	182	········{
183	············"cell_type":·"code",	183	············"cell_type":·"code",
184	············"execution_count":·null,	184	············"execution_count":·null,
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1	{	1	{
2	····"cells":·[	2	····"cells":·[
3	········{	3	········{
4	············"cell_type":·"markdown",	4	············"cell_type":·"markdown",
5	············"id":·"~~7d477~~c33",	5	············"id":·"a39d5c1a",
6	············"metadata":·{},	6	············"metadata":·{},
7	············"source":·[	7	············"source":·[
8	················"#·Mixed·formulation·for·the·Poisson·equation"	8	················"#·Mixed·formulation·for·the·Poisson·equation"
9	············]	9	············]
10	········},	10	········},
11	········{	11	········{
12	············"cell_type":·"markdown",	12	············"cell_type":·"markdown",
13	············"id":·"~~253cca59~~",	13	············"id":·"3fb5786c",
14	············"metadata":·{},	14	············"metadata":·{},
15	············"source":·[	15	············"source":·[
16	················"This·demo·illustrates·how·to·solve·Poisson·equation·using·a·mixed\n",	16	················"This·demo·illustrates·how·to·solve·Poisson·equation·using·a·mixed\n",
17	················"(two-field)·formulation.·In·particular,·it·illustrates·how·to\n",	17	················"(two-field)·formulation.·In·particular,·it·illustrates·how·to\n",
18	················"\n",	18	················"\n",
19	················"*·Use·mixed·and·non-continuous·finite·element·spaces.\n",	19	················"*·Use·mixed·and·non-continuous·finite·element·spaces.\n",
20	················"*·Set·essential·boundary·conditions·for·subspaces·and·$H(\\mathrm{div})$·spaces.\n"	20	················"*·Set·essential·boundary·conditions·for·subspaces·and·$H(\\mathrm{div})$·spaces.\n"
21	············]	21	············]
22	········},	22	········},
23	········{	23	········{
24	············"cell_type":·"markdown",	24	············"cell_type":·"markdown",
25	············"id":·"~~7e74~~5~~8ca~~",	25	············"id":·"caa0551c",
26	············"metadata":·{},	26	············"metadata":·{},
27	············"source":·[	27	············"source":·[
28	················"```{admonition}·Download·sources\n",	28	················"```{admonition}·Download·sources\n",
29	················":class:·download\n",	29	················":class:·download\n",
30	················"\n",	30	················"\n",
31	················"*·{download}`Python·script·<./demo_mixed-poisson.py>`\n",	31	················"*·{download}`Python·script·<./demo_mixed-poisson.py>`\n",
32	················"*·{download}`Jupyter·notebook·<./demo_mixed-poisson.ipynb>`\n",	32	················"*·{download}`Jupyter·notebook·<./demo_mixed-poisson.ipynb>`\n",
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99	················"\n",	99	················"\n",
100	················"##·Implementation"	100	················"##·Implementation"
101	············]	101	············]
102	········},	102	········},
103	········{	103	········{
104	············"cell_type":·"code",	104	············"cell_type":·"code",
105	············"execution_count":·null,	105	············"execution_count":·null,
106	············"id":·"~~1f62a631~~",	106	············"id":·"ae4eec0b",
107	············"metadata":·{},	107	············"metadata":·{},
108	············"outputs":·[],	108	············"outputs":·[],
109	············"source":·[	109	············"source":·[
110	················"\n",	110	················"\n",
111	················"from·mpi4py·import·MPI\n",	111	················"from·mpi4py·import·MPI\n",
112	················"from·petsc4py·import·PETSc\n",	112	················"from·petsc4py·import·PETSc\n",
113	················"\n",	113	················"\n",


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