From f13c961ab323c30e42e1cd59a78c48f48faf48b9 Mon Sep 17 00:00:00 2001 From: Anurag Rao Date: Tue, 2 Sep 2025 05:40:59 -0700 Subject: [PATCH 1/7] added demo for adaptive multigrid --- .../adaptive_convergence_9.png | Bin 0 -> 34541 bytes demos/adaptive_multigrid/demo.py.rst | 195 ++++++++++++++++++ demos/adaptive_multigrid/eta_l15.png | Bin 0 -> 338521 bytes demos/adaptive_multigrid/eta_l3.png | Bin 0 -> 47146 bytes demos/adaptive_multigrid/eta_l6.png | Bin 0 -> 86368 bytes demos/adaptive_multigrid/initial_mesh.png | Bin 0 -> 44940 bytes demos/adaptive_multigrid/solution_l1.png | Bin 0 -> 32367 bytes demos/adaptive_multigrid/solution_l15.png | Bin 0 -> 155993 bytes demos/adaptive_multigrid/solution_l4.png | Bin 0 -> 104185 bytes 9 files changed, 195 insertions(+) create mode 100644 demos/adaptive_multigrid/adaptive_convergence_9.png create mode 100644 demos/adaptive_multigrid/demo.py.rst create mode 100644 demos/adaptive_multigrid/eta_l15.png create mode 100644 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+================================================= + +The purpose of this demo is to show how to use Firedrake's multigrid solver on a hierarchy of adaptively refined Netgen meshes. +We will first have a look at how to use the AdaptiveMeshHierarchy to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L shaped domain. +Finally, we will show how to use the AdaptiveMeshHierarchy and AdaptiveTransferManager as arguments to Firedrake solvers. The AdaptiveMeshHierarchy contains information of the mesh hierarchy and the parent child relations between the meshes. +The AdaptiveTransferManager deals with the transfer operator logic across any given levels in the hierarchy. +We begin by importing the necessary libraries :: + + from firedrake import * + from netgen.occ import * + from ngsPETSc import AdaptiveMeshHierarchy, AdaptiveTransferManager + +Constructing the Mesh Hierarchy +--------------------------- +We first must construct the domain over which we will solve the problem. For a more comprehensive demo on how to use Open Cascade Technology (OCC) and Constructive Solid Geometry (CSG), see `Netgen integration in Firedrake `_ . +We begin with the L-shaped domain, which we build as the union of two rectangles: :: + rect1 = WorkPlane(Axes((0,0,0), n=Z, h=X)).Rectangle(1,2).Face() + rect2 = WorkPlane(Axes((0,1,0), n=Z, h=X)).Rectangle(2,1).Face() + L = rect1 + rect2 + + geo = OCCGeometry(L, dim=2) + ngmsh = geo.GenerateMesh(maxh=0.5) + mesh = Mesh(ngmsh) + +It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the AdaptiveMeshHierarchy. To call the constructor to the hierarchy, we must pass the initial mesh as a list. Our initial mesh looks like this: + +.. figure:: initial_mesh.png + :align: center + :alt: Initial mesh. + +We will also initialize the AdaptiveTransferManager here: :: + + amh = AdaptiveMeshHierarchy([mesh]) + atm = AdaptiveTransferManager() + +Poisson Problem +------------------------- +Now we can define a simple Poisson Problem + +.. math:: + + - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega + +Our approach strongly follows the similar problem in this `lecture course `_. We define the function solve_poisson. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the DirichletBC object and apply it to the entire boundary: :: + + def solve_poisson(mesh, params): + V = FunctionSpace(mesh, "CG", 1) + uh = Function(V, name="Solution") + v = TestFunction(V) + bc = DirichletBC(V, Constant(0), "on_boundary") + f = Constant(1) + F = inner(grad(uh), grad(v))*dx - inner(f, v)*dx + + problem = NonlinearVariationalProblem(F, uh, bc) + solver = NonlinearVariationalSolver(problem, solver_parameters=params) + solver.set_transfer_manager(atm) + solver.solve() + return uh + +Note the code after the construction of the NonlinearVariationalProblem(). To use the AdaptiveMeshHierarchy with the existing Firedrake solver, we have to set the AdaptiveTransferManager as the transfer manager of the multigrid solver. +For the parameters of the multigrid solver, we will be using patch relaxation, which we define with :: + solver_params = { + "mat_type": "matfree", + "ksp_type": "cg", + "pc_type": "mg", + "mg_levels": { + "ksp_type": "chebyshev", + "ksp_max_it": 1, + "pc_type": "python", + "pc_python_type": "firedrake.PatchPC", + "patch": { + "pc_patch": { + "construct_type": "star", + "construct_dim": 0, + "sub_mat_type": "seqdense", + "dense_inverse": True, + "save_operators": True, + "precompute_element_tensors": True, + }, + "sub_ksp_type": "preonly", + "sub_pc_type": "lu", + }, + }, + "mg_coarse": { + "ksp_type": "preonly", + "pc_type": "python", + "pc_python_type": "firedrake.AssembledPC", + "assembled": {"ksp_type": "preonly", "pc_type": "lu"}, + }, + } + +For more information about patch relaxation, see `Using patch relaxation for multigrid `_. The initial solution is shown below + +.. figure:: solution_l1.png + :align: center + :alt: Initial Solution from multigrid with initial mesh. + + +Adaptive Mesh Refinement +------------------------- +In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the AdaptiveMeshHierarchy. +For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element: + +.. math:: + \eta_K^2 = h_K^2 \int_K | f + \nabla^2 u_h |^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s, + +where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`\llbracket \cdot \rrbracket` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. We can use a trick to compute the estimator on each element. We transform the above estimator into the variational problem + +.. math:: + \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s w \mathrm{d}x + +Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space. :: + + def estimate_error(mesh, uh): + W = FunctionSpace(mesh, "DG", 0) + eta_sq = Function(W) + w = TestFunction(W) + f = Constant(1) + h = CellDiameter(mesh) # symbols for mesh quantities + n = FacetNormal(mesh) + v = CellVolume(mesh) + + G = ( # compute cellwise error estimator + inner(eta_sq / v, w)*dx + - inner(h**2 * (f + div(grad(uh)))**2, w) * dx + - inner(h('+')/2 * jump(grad(uh), n)**2, w('+')) * dS + - inner(h('-')/2 * jump(grad(uh), n)**2, w('-')) * dS + ) + + sp = {"mat_type": "matfree", "ksp_type": "richardson", "pc_type": "jacobi"} + solve(G == 0, eta_sq, solver_parameters=sp) + eta = Function(W).interpolate(sqrt(eta_sq)) # compute eta from eta^2 + + with eta.dat.vec_ro as eta_: # compute estimate for error in energy norm + error_est = sqrt(eta_.dot(eta_)) + return (eta, error_est) + +The next step is to choose which elements to refine. For this we Dörfler marking, developed by Professor Willy Dörfler: + +.. math:: + \eta_K \geq \theta \text{max}_L \eta_L + +The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`theta=0.5`. +With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid with patch relaxation up till level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we refine the mesh and repeat. :: + + max_iterations = 15 + error_estimators = [] + dofs = [] + for i in range(max_iterations): + print(f"level {i}") + + uh = solve_poisson(mesh, patch_relax) + VTKFile(f"output/poisson_l/{max_iterations}/adaptive_loop_{i}.pvd").write(uh) + + (eta, error_est) = estimate_error(mesh, uh) + VTKFile(f"output/poisson_l/{max_iterations}/eta_{i}.pvd").write(eta) + + print(f" ||u - u_h|| <= C x {error_est}") + error_estimators.append(error_est) + dofs.append(uh.function_space().dim()) + + if i != max_iterations - 1: + amh.adapt(eta, theta) + +To perform Dörfler marking, refine the current mesh, and add the mesh to the AdaptiveMeshHierarchy, we us the amh.adapt(eta, theta) method. In this method the input is the recently computed error estimator :math:`eta` and the Dörfler marking parameter :math:`theta`. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy. This is the amh.add_mesh(mesh). In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy. +The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area. + ++-------------------------------+-------------------------------+-------------------------------+ +| .. figure:: eta_l3.png | .. figure:: eta_l6.png | .. figure:: eta_l15.png | +| :align: center | :align: center | :align: center | +| :height: 250px | :height: 250px | :height: 250px | +| :alt: Eta at level 3 | :alt: Eta at level 6 | :alt: Eta at level 15 | +| | | | +| *Level 3* | *Level 6* | *Level 15* | ++-------------------------------+-------------------------------+-------------------------------+ + +The solutions at level 4 and 15 are shown below. + ++------------------------------------+------------------------------------+ +| .. figure:: solution_l4.png | .. figure:: solution_l15.png | +| :align: center | :align: center | +| :height: 300px | :height: 300px | +| :alt: Solution, level 4 | :alt: Solution, level 15 | +| | | +| *MG solution at level 4* | *MG solution at level 15* | ++------------------------------------+------------------------------------+ + + +The convergence follows the expected behavior: + +.. figure:: adaptive_convergence_9.png + :align: center + :alt: Convergence of the error estimator. + diff --git a/demos/adaptive_multigrid/eta_l15.png b/demos/adaptive_multigrid/eta_l15.png new file mode 100644 index 0000000000000000000000000000000000000000..05efad50c317c5f000b21c6e41f11121009c188a GIT binary patch literal 338521 zcmc$FWmKF^ljz{?B)Gcb03cUBhAcen2PTOyZUX>-(6$yASCJJL zCslE>x3C780|2z~u5qHY(2m$c-R2xC_j#I*HJpQ@x^vp2l9f~o#Kn?~zrbM5a*N^3 z?^2-FHeP<({jK5_o~^y3ss>agrKxWeO@H3_(+sXp?)-dxJ0I@+Kg(K9Bmj~?sg@Sh z$@^zN$c4qFxK}X5^|=XC78*Eot^_m7y<+3i+=EQC27M5P?~BpkiVNj`jrHyueOC{; zff>$w5+&lsbq`Pxu5CJJ>*XKF&Jx48MB?%l`kZz1RB)j)QWS0wSLsJ!=-3q%7 z+Dd3N_c2F5rhxHwG(1`moQVtl0UgR3qoeS;tGLif2lCHHj&V(+?F$K{q(lCUIN)3Z zG=_IHN#-iU{UrSM+AhLS4T(jay@J%hxG%H{(>4WFX^$K}=Eo3q?f${Ra#WfC#e& zKz~3G9|Qkm002-4p#Zp#Z_JM&kq7x-RLGS)sQ-p#|6vqW7nl79P&ailH@9=PvUhoe zsH6K(wP>xO?V_!yz-MZ2%M3KLH!)}Suyy!H1R&tS_W|0Py8uZ&Y(aL;d>(@2|G~lc z0sn*kLQeW0EG{;JNAEX*wALWrcKqykQ67JOW@(U|3FYgx?wl8dKOdlLf&YpHIKo2H6XNvz+@?Ux+%$-f0tQ}me z?d?ea(F-)OcXbgYC;!LLe}De-cba=x|Bod*=YLJ>V}f7)p?qOwX8H2px<5n({=xF8 zSbLa*bR?{8Kiu01yGlN{DKBK%8}2 zCQ^;MQhDSqvNh6zsoL7hbh#Ty0fD3tnB$-7EuO7~D=Wco5$oNx?VAVvX_e8FQ5b}Q z;*gZy07$5sSzS#n*U^Q)oj$r3yi2=XQBe~WWfx`OMmz6jdZXaTtMIGv7QdIryVLyd zchzqX`BCQjP`miJOFb>HL!2yC0gfn9Hpc&;JkIbRZ|A#H`rhwDQ?UHb=7wPYMfqRM z(%+1c8)7v^c})JFx?+u>O!HnJU3r*+|4;h=Z*fvD$^&N!g)Cn=ulS1K0@=}#Di9|k z+Y#5#bs{-hjwCl>YUig6%0$&;ltRpy;Gf*^@*J4)Rs!V5k0M>2@6HG~<8w??Te$Hc zS4r52B#j#+FUhA@&f@{Ky2PyHUSrx?LvgY7s@rFAFls90iv4KbQnH~vE&q8Ga_x8L z*x~!1hL`dZm)J1lDxpvVJ3Kkk|tH#mIbJNSlB@9&z^>b|BL``TJel(!{ z*i25v@BXY2AmQ+~NS=^9NXOeWUvYesfW2fnYm+H63B7tquBu|NRlT_!c2?e%F6tm#;72@H#oBTccNV|o;m0I^#~7KHfXjw zrFtY2eYiX!+pUv`J2h+lK$R07)wHgpyt!)l>-$;L?>=Ygr(cJmJl{2@jP)*m{ZxyU z<-t$1{CvW;^rq1rE2U@V$-=byJ8o@dVhnH?wVP?xe03nP!p54OF}9x?gV+iDl6r;oW{M|v#u7qWJf0F+OU*nTeNRSyQ+1wcY+Mzx*Na2 z8eErRXLVw8%H)B&Y!ywGu8`+&3n#YhN?BHt-6n*WWOkYv#7+ydo1BVMyS)SO-q8S0Ab~U77B@ zCIZ*$OpJrNTX?|B#v{G%u7skTMQk8$@tm^&1x#7b;hRLcFr_WcelBF0PTvsoIEX6N zYvJ`AOOA*YXzZk%ybc`5TlPS~&(S8#P6WzXdZ{)G5+)k9(mfv<4Y=bEv4`bUGmaW= zb9C*8FRYSzmj6xakm-aJtltduE7wn|#;%fw-Nx^KPNl-Y@g%#8KfyoAao3$5i5x{k z>&v#>3keSCzh>_HGPJ4XyXar#)9I?%mEom)mmAWZ;vs%#iNkki>xr;wA!8WRRXD8j ztBYoOJW@$X1S-C_$lv!p565n$`^3R$qGY0h2k4 zNRDeI{&yKtD!PSNJwZP=L5fLntbO+8y~_OwkgkYtCe}l0CgM4oWn(htf=0|1>~Bbr z7cUn&YlxzIec#D;d!kaZDP|Ru$CDS-_xIv@4-^Z&uPr^HKsiAGf-nwl!t+!=LRIX1 zKFXe_XORpw!Tg~cwJX6iDI-=LO}c|#!p+TGFmcYwrHs9e`dihzv=tUV=CJ#?&FL=x zw*tZSlA(NQEXvxo-41}V9TfOJAU0(RB0=|f%NL1-R z6~6B}UQ)2SDq(`N7QO*2UXunON=^X`5ejnnmHCudth~NWw4?e;JI4(YJ9H5{K}gaN zN`HP*^$WHNbmVWb6UNFia>P0OqRaPFUI8P>ZtX_8>0#pC7A2>n_ZZW{(-Qi`)tW&b z%F6H&CDYNCE3ig)cO;F|;XyuXod$*q1WGyk#}bMl4fv$3eM`u^ZA4zs5 zqiqjgTQ%-t!Nw4Q>2W2=ph1pShjs9NW8+X3Kr)v@hZ#gyl-0sivkh)o;~3lb9NRD2 zb07kX>3$2jaq_&1=rH=5Baui4o)Dlq^y1#nejynpB}`3T_S89BPiY3!m8#m0dg>f) zCUUcHE?%8)_EOz}qye_tny}7nSSxW$&tGWs>Y$eMI1`P`zgNUQ8(L~wQ0QH2US_0tHght~&l35cS#tO?a4@y1C~inmNafcV zoGXmEfRs0E4Hi*HzFCSF3dND7_jE}2rRyOus+dPN`eBz>=<9J~*8 zb4U^UWk#)+D)pu6hcBjaUDi2NuxM|X37hH`-5YJosZc2yte@rZ>RPp5fKj6kU+)_xF@6~wJuk!MRk(KYZy*34iu?O$uvfUs+)KyC$L2o{bc!? 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The AdaptiveMeshHierarchy contains information of the mesh hierarchy and the parent child relations between the meshes. -The AdaptiveTransferManager deals with the transfer operator logic across any given levels in the hierarchy. +We will first have a look at how to use the *AdaptiveMeshHierarchy* to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain. +Finally, we will show how to use the *AdaptiveMeshHierarchy* and *AdaptiveTransferManager* as arguments to Firedrake solvers. The *AdaptiveMeshHierarchy* contains information of the mesh hierarchy and the parent child relations between the meshes. +The *AdaptiveTransferManager* deals with the transfer operator logic across any given levels in the hierarchy. We begin by importing the necessary libraries :: from firedrake import * @@ -23,26 +23,26 @@ We begin with the L-shaped domain, which we build as the union of two rectangles ngmsh = geo.GenerateMesh(maxh=0.5) mesh = Mesh(ngmsh) -It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the AdaptiveMeshHierarchy. To call the constructor to the hierarchy, we must pass the initial mesh as a list. Our initial mesh looks like this: +It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the *AdaptiveMeshHierarchy*. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this: .. figure:: initial_mesh.png :align: center :alt: Initial mesh. -We will also initialize the AdaptiveTransferManager here: :: +We will also initialize the *AdaptiveTransferManager* here: :: - amh = AdaptiveMeshHierarchy([mesh]) + amh = AdaptiveMeshHierarchy(mesh) atm = AdaptiveTransferManager() Poisson Problem ------------------------- -Now we can define a simple Poisson Problem +Now we can define a simple Poisson problem .. math:: - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega -Our approach strongly follows the similar problem in this `lecture course `_. We define the function solve_poisson. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the DirichletBC object and apply it to the entire boundary: :: +Our approach strongly follows the similar problem in this `lecture course `_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the *DirichletBC* object and apply it to the entire boundary: :: def solve_poisson(mesh, params): V = FunctionSpace(mesh, "CG", 1) @@ -58,7 +58,7 @@ Our approach strongly follows the similar problem in this `lecture course `_. The initial solution is shown below +For more information about patch relaxation, see `Using patch relaxation for multigrid `_. The initial solution is shown below. .. figure:: solution_l1.png :align: center @@ -99,13 +99,13 @@ For more information about patch relaxation, see `Using patch relaxation for mul Adaptive Mesh Refinement ------------------------- -In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the AdaptiveMeshHierarchy. +In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the *AdaptiveMeshHierarchy*. For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element: .. math:: \eta_K^2 = h_K^2 \int_K | f + \nabla^2 u_h |^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s, -where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`\llbracket \cdot \rrbracket` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. We can use a trick to compute the estimator on each element. We transform the above estimator into the variational problem +where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`\llbracket \cdot \rrbracket` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem .. math:: \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s w \mathrm{d}x @@ -136,7 +136,7 @@ Our approach will be to compute the estimator over all elements and selectively error_est = sqrt(eta_.dot(eta_)) return (eta, error_est) -The next step is to choose which elements to refine. For this we Dörfler marking, developed by Professor Willy Dörfler: +The next step is to choose which elements to refine. For this we Dörfler marking [Dörfler1996]: .. math:: \eta_K \geq \theta \text{max}_L \eta_L @@ -163,7 +163,7 @@ With these helper functions complete, we can solve the system iteratively. In th if i != max_iterations - 1: amh.adapt(eta, theta) -To perform Dörfler marking, refine the current mesh, and add the mesh to the AdaptiveMeshHierarchy, we us the amh.adapt(eta, theta) method. In this method the input is the recently computed error estimator :math:`eta` and the Dörfler marking parameter :math:`theta`. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy. This is the amh.add_mesh(mesh). In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy. +To perform Dörfler marking, refine the current mesh, and add the mesh to the *AdaptiveMeshHierarchy*, we us the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator :math:`eta` and the Dörfler marking parameter :math:`theta`. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy. The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area. +-------------------------------+-------------------------------+-------------------------------+ @@ -187,9 +187,13 @@ The solutions at level 4 and 15 are shown below. +------------------------------------+------------------------------------+ -The convergence follows the expected behavior: +The convergence follows the expected optimal behavior: .. figure:: adaptive_convergence_9.png :align: center :alt: Convergence of the error estimator. +References +------------------------- +[Dörfler1996] W. Dörfler. A convergent adaptive algorithm for poisson’s equation. SIAM Journal on Numerical Analysis, 33(3):1106–1124, 1996. + From 2d3cabbcba71d161990cff351ba66f94c78c65ee Mon Sep 17 00:00:00 2001 From: Anurag Rao Date: Tue, 2 Sep 2025 08:47:26 -0700 Subject: [PATCH 3/7] fixed type by adding 'use' --- demos/adaptive_multigrid/demo.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/adaptive_multigrid/demo.py.rst b/demos/adaptive_multigrid/demo.py.rst index 6705a86c5e..6e305570f4 100644 --- a/demos/adaptive_multigrid/demo.py.rst +++ b/demos/adaptive_multigrid/demo.py.rst @@ -136,7 +136,7 @@ Our approach will be to compute the estimator over all elements and selectively error_est = sqrt(eta_.dot(eta_)) return (eta, error_est) -The next step is to choose which elements to refine. For this we Dörfler marking [Dörfler1996]: +The next step is to choose which elements to refine. For this we use Dörfler marking [Dörfler1996]: .. math:: \eta_K \geq \theta \text{max}_L \eta_L From b88fb70dbff0f01fc24eb78ebb5104fe3e731351 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Tue, 27 Jan 2026 14:35:24 +0000 Subject: [PATCH 4/7] Test demo --- .../{demo.py.rst => adaptive_multigrid.py.rst} | 0 docs/source/advanced_tut.rst | 1 + tests/firedrake/demos/test_demos_run.py | 1 + 3 files changed, 2 insertions(+) rename demos/adaptive_multigrid/{demo.py.rst => adaptive_multigrid.py.rst} (100%) diff --git a/demos/adaptive_multigrid/demo.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst similarity index 100% rename from demos/adaptive_multigrid/demo.py.rst rename to demos/adaptive_multigrid/adaptive_multigrid.py.rst diff --git a/docs/source/advanced_tut.rst b/docs/source/advanced_tut.rst index 5fa8f360ed..084c605609 100644 --- a/docs/source/advanced_tut.rst +++ b/docs/source/advanced_tut.rst @@ -26,6 +26,7 @@ element systems. Full-waveform inversion: spatial and wave sources parallelism. 1D Vlasov-Poisson equation using vertical independent function spaces. Degree-independent multigrid convergence using patch relaxation. + Multigrid on adaptively-refined mesh hierarchies. Monolithic multigrid with Vanka relaxation for Stokes. Vertex/edge star multigrid relaxation for H(div). Auxiliary space patch relaxation multigrid for H(curl). diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py index daaadc4e15..297f95a459 100644 --- a/tests/firedrake/demos/test_demos_run.py +++ b/tests/firedrake/demos/test_demos_run.py @@ -18,6 +18,7 @@ DEMO_DIR = join(CWD, "..", "..", "..", "demos") SERIAL_DEMOS = [ + Demo(("adaptive_multigrid", "adaptive_multigrid"), ["netgen"]), Demo(("benney_luke", "benney_luke"), ["vtk"]), Demo(("boussinesq", "boussinesq"), []), Demo(("burgers", "burgers"), ["vtk"]), From d083087acf6da4df8dd34a7de100f3a91051c480 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Tue, 27 Jan 2026 16:08:28 +0000 Subject: [PATCH 5/7] links --- .../adaptive_convergence_9.png | Bin 34541 -> 0 bytes .../adaptive_multigrid.py.rst | 199 ++++++++++-------- demos/demo_references.bib | 11 + 3 files changed, 119 insertions(+), 91 deletions(-) delete mode 100644 demos/adaptive_multigrid/adaptive_convergence_9.png diff --git 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zVc)3*5ece*igJLpCq6~S`9(LxymyxtXvvT%5-mEz7Nzy6rqPo{m@As3zg6e-qL6hg z$yDX6nCC0PQbf(^@mVGTJ(osL5$8sFN1>!>ZX^0@!aanw+7*G|wN|ob&Daz3fPn>^ zE=BRS=Ka&&2vEB#p3=-k_Ps5>#--=+zyo(?yKyvWrc`m`zvHY;?xlE1g)ka&o48%iGL%WAn0Q=@;HEPNoMBK&~%G&cA& z6fH3+UK~l?dYDfPv1dB`$^5@2?Nx~m3bv?DbgVxvn7i$G1(EtDd diff --git a/demos/adaptive_multigrid/adaptive_multigrid.py.rst b/demos/adaptive_multigrid/adaptive_multigrid.py.rst index 6e305570f4..43bb2e0a9a 100644 --- a/demos/adaptive_multigrid/adaptive_multigrid.py.rst +++ b/demos/adaptive_multigrid/adaptive_multigrid.py.rst @@ -1,54 +1,59 @@ -Adaptive Multigrid Methods using the AdaptiveMeshHierarchy -================================================= +Adaptive Multigrid Methods using AdaptiveMeshHierarchy +====================================================== + + +Contributed by Anurag Rao The purpose of this demo is to show how to use Firedrake's multigrid solver on a hierarchy of adaptively refined Netgen meshes. -We will first have a look at how to use the *AdaptiveMeshHierarchy* to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain. -Finally, we will show how to use the *AdaptiveMeshHierarchy* and *AdaptiveTransferManager* as arguments to Firedrake solvers. The *AdaptiveMeshHierarchy* contains information of the mesh hierarchy and the parent child relations between the meshes. -The *AdaptiveTransferManager* deals with the transfer operator logic across any given levels in the hierarchy. +We will first have a look at how to use the :class:`.AdaptiveMeshHierarchy` to construct the mesh hierarchy with Netgen meshes, then we will consider a solution to the Poisson problem on an L-shaped domain. +Finally, we will show how to use the :class:`.AdaptiveMeshHierarchy` and :class:`.AdaptiveTransferManager` to construct a scalable solver. The :class:`.AdaptiveMeshHierarchy` contains information of the mesh hierarchy and the parent child relations between the meshes. +The :class:`.AdaptiveTransferManager` deals with the transfer operator logic across any given levels in the hierarchy. We begin by importing the necessary libraries :: from firedrake import * from netgen.occ import * - from ngsPETSc import AdaptiveMeshHierarchy, AdaptiveTransferManager + import numpy Constructing the Mesh Hierarchy ---------------------------- -We first must construct the domain over which we will solve the problem. For a more comprehensive demo on how to use Open Cascade Technology (OCC) and Constructive Solid Geometry (CSG), see `Netgen integration in Firedrake `_ . +------------------------------- +We first must construct the domain over which we will solve the problem. For a more comprehensive demo on how to use Open Cascade Technology (OCC) and Constructive Solid Geometry (CSG), +see `Netgen integration in Firedrake `_. We begin with the L-shaped domain, which we build as the union of two rectangles: :: - rect1 = WorkPlane(Axes((0,0,0), n=Z, h=X)).Rectangle(1,2).Face() - rect2 = WorkPlane(Axes((0,1,0), n=Z, h=X)).Rectangle(2,1).Face() - L = rect1 + rect2 - geo = OCCGeometry(L, dim=2) - ngmsh = geo.GenerateMesh(maxh=0.5) - mesh = Mesh(ngmsh) + rect1 = WorkPlane(Axes((0,0,0), n=Z, h=X)).Rectangle(1,2).Face() + rect2 = WorkPlane(Axes((0,1,0), n=Z, h=X)).Rectangle(2,1).Face() + L = rect1 + rect2 + + geo = OCCGeometry(L, dim=2) + ngmsh = geo.GenerateMesh(maxh=0.5) + mesh = Mesh(ngmsh) -It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the *AdaptiveMeshHierarchy*. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this: +It is important to convert the initial Netgen mesh into a Firedrake mesh before constructing the :class:`.AdaptiveMeshHierarchy`. To call the constructor to the hierarchy, we must pass the initial mesh. Our initial mesh looks like this: .. figure:: initial_mesh.png :align: center :alt: Initial mesh. -We will also initialize the *AdaptiveTransferManager* here: :: +We will also initialize the :class:`.AdaptiveTransferManager` here: :: - amh = AdaptiveMeshHierarchy(mesh) - atm = AdaptiveTransferManager() + amh = AdaptiveMeshHierarchy(mesh) + atm = AdaptiveTransferManager() Poisson Problem -------------------------- +--------------- Now we can define a simple Poisson problem .. math:: - \nabla^2 u = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega -Our approach strongly follows the similar problem in this `lecture course `_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the *DirichletBC* object and apply it to the entire boundary: :: +Our approach strongly follows the similar problem in this `lecture course `_. We define the function ``solve_poisson``. The first lines correspond to finding a solution in the CG1 space. The variational problem is formulated with F, where f is the constant function equal to 1. Since we want Dirichlet boundary conditions, we construct the :class:`.DirichletBC` object and apply it to the entire boundary: :: def solve_poisson(mesh, params): V = FunctionSpace(mesh, "CG", 1) - uh = Function(V, name="Solution") + uh = Function(V, name="solution") v = TestFunction(V) - bc = DirichletBC(V, Constant(0), "on_boundary") + bc = DirichletBC(V, 0, "on_boundary") f = Constant(1) F = inner(grad(uh), grad(v))*dx - inner(f, v)*dx @@ -58,39 +63,29 @@ Our approach strongly follows the similar problem in this `lecture course `_. The initial solution is shown below. + "mat_type": "matfree", + "ksp_type": "cg", + "pc_type": "mg", + "mg_levels": { + "ksp_type": "chebyshev", + "ksp_max_it": 1, + "pc_type": "jacobi", + }, + "mg_coarse": { + "mat_type": "aij", + "pc_type": "lu", + }, + } + +Alternatively for high-order CG elements, it is recommended to use patch relaxation +to achieve degree-independent multigrid convergence. +For more information +see :doc:`Using patch relaxation for multigrid `. +The initial solution is shown below. .. figure:: solution_l1.png :align: center @@ -98,17 +93,17 @@ For more information about patch relaxation, see `Using patch relaxation for mul Adaptive Mesh Refinement -------------------------- -In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the *AdaptiveMeshHierarchy*. +------------------------ +In this section we will discuss how to adaptively refine select elements and add the newly refined mesh into the :class:`.AdaptiveMeshHierarchy`. For this problem, we will be using the Babuška-Rheinbolt a-posteriori estimate for an element: .. math:: - \eta_K^2 = h_K^2 \int_K | f + \nabla^2 u_h |^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s, + \eta_K^2 = h_K^2 \int_K \| f + \nabla^2 u_h \|^2 \mathrm{d}x + \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s, -where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`\llbracket \cdot \rrbracket` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem +where :math:`K` is the element, :math:`h_K` is the diameter of the element, :math:`n` is the normal, and :math:`⟦ \cdot ⟧` is the jump operator. The a-posteriori estimator is computed using the solution at the current level :math:`h`. Integrating over the domain and using the fact that the components of the estimator are piecewise constant on each cell, we can transform the above estimator into the variational problem .. math:: - \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} \llbracket \nabla u_h \cdot n \rrbracket^2 \mathrm{d}s w \mathrm{d}x + \int_\Omega \eta_K^2 w \mathrm{d}x = \int_\Omega \sum_K h_K^2 \int_K (f + \text{div} (\text{grad} u_h) )^2 \mathrm{d}x w \mathrm{d}x + \int_\Omega \sum_K \frac{h_K}{2} \int_{\partial K \setminus \partial \Omega} ⟦ \nabla u_h \cdot n ⟧^2 \mathrm{d}s w \mathrm{d}x Our approach will be to compute the estimator over all elements and selectively choose to refine only those that contribute most to the error. To compute the error estimator, we use the function below to solve the variational formulation of the error estimator. Since our estimator is a constant per element, we use a DG0 function space. :: @@ -117,54 +112,73 @@ Our approach will be to compute the estimator over all elements and selectively eta_sq = Function(W) w = TestFunction(W) f = Constant(1) - h = CellDiameter(mesh) # symbols for mesh quantities + + # symbols for mesh quantities + h = CellDiameter(mesh) n = FacetNormal(mesh) v = CellVolume(mesh) - G = ( # compute cellwise error estimator - inner(eta_sq / v, w)*dx - - inner(h**2 * (f + div(grad(uh)))**2, w) * dx - - inner(h('+')/2 * jump(grad(uh), n)**2, w('+')) * dS - - inner(h('-')/2 * jump(grad(uh), n)**2, w('-')) * dS - ) + # compute cellwise error estimator + G = (inner(eta_sq / v, w)*dx + - inner(h**2 * (f + div(grad(uh)))**2, w) * dx + - inner(h('+')/2 * jump(grad(uh), n)**2, w('+')) * dS + - inner(h('-')/2 * jump(grad(uh), n)**2, w('-')) * dS + ) sp = {"mat_type": "matfree", "ksp_type": "richardson", "pc_type": "jacobi"} solve(G == 0, eta_sq, solver_parameters=sp) eta = Function(W).interpolate(sqrt(eta_sq)) # compute eta from eta^2 with eta.dat.vec_ro as eta_: # compute estimate for error in energy norm - error_est = sqrt(eta_.dot(eta_)) - return (eta, error_est) + error_est = eta_.norm() + return eta, error_est -The next step is to choose which elements to refine. For this we use Dörfler marking [Dörfler1996]: +The next step is to choose which elements to refine. For this we use Dörfler marking :cite:`Dorfler1996`: .. math:: \eta_K \geq \theta \text{max}_L \eta_L -The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`theta=0.5`. +The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`\theta=0.5`. With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid with patch relaxation up till level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we refine the mesh and repeat. :: + theta = 0.5 max_iterations = 15 - error_estimators = [] + est_errors = [] dofs = [] for i in range(max_iterations): - print(f"level {i}") - - uh = solve_poisson(mesh, patch_relax) - VTKFile(f"output/poisson_l/{max_iterations}/adaptive_loop_{i}.pvd").write(uh) - - (eta, error_est) = estimate_error(mesh, uh) - VTKFile(f"output/poisson_l/{max_iterations}/eta_{i}.pvd").write(eta) - - print(f" ||u - u_h|| <= C x {error_est}") - error_estimators.append(error_est) - dofs.append(uh.function_space().dim()) - - if i != max_iterations - 1: - amh.adapt(eta, theta) + print(f"level {i}") + + mesh = amh[-1] + uh = solve_poisson(mesh, solver_params) + VTKFile(f"output/adaptive_loop_{i}.pvd").write(uh) -To perform Dörfler marking, refine the current mesh, and add the mesh to the *AdaptiveMeshHierarchy*, we us the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator :math:`eta` and the Dörfler marking parameter :math:`theta`. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy. -The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area. + (eta, error_est) = estimate_error(mesh, uh) + VTKFile(f"output/eta_{i}.pvd").write(eta) + + est_errors.append(error_est) + dofs.append(uh.function_space().dim()) + + print(f" ||u - u_h|| <= C * {error_est}") + if len(dofs) > 1: + rate = -numpy.diff(numpy.log(est_errors)) / numpy.diff(numpy.log(dofs)) + print(f" rate = {rate[-1]}") + + if i != max_iterations - 1: + amh.adapt(eta, theta) + + from matplotlib import pyplot as plt + + opt_errors = est_errors[0] * (numpy.array(dofs)/dofs[0]) ** -0.5 + plt.loglog(dofs, est_errors, '-o', markersize = 3, label="Estimated error") + plt.loglog(dofs, opt_errors, '--', markersize = 3, label="Optimal convergence") + plt.ylabel("Error estimate of the energy norm") + plt.xlabel("Number of degrees of freedom") + plt.legend() + plt.savefig("output/adaptive_convergence.png") + + +To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy. +The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L-shape at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area. +-------------------------------+-------------------------------+-------------------------------+ | .. figure:: eta_l3.png | .. figure:: eta_l6.png | .. figure:: eta_l15.png | @@ -189,11 +203,14 @@ The solutions at level 4 and 15 are shown below. The convergence follows the expected optimal behavior: -.. figure:: adaptive_convergence_9.png +.. figure:: adaptive_convergence.png :align: center :alt: Convergence of the error estimator. -References -------------------------- -[Dörfler1996] W. Dörfler. A convergent adaptive algorithm for poisson’s equation. SIAM Journal on Numerical Analysis, 33(3):1106–1124, 1996. +A runnable python version of this demo can be found :demo:`here`. + +.. rubric:: References + +.. bibliography:: demo_references.bib + :filter: docname in docnames diff --git a/demos/demo_references.bib b/demos/demo_references.bib index 31c360853f..3a6611f295 100644 --- a/demos/demo_references.bib +++ b/demos/demo_references.bib @@ -357,3 +357,14 @@ @article{Brubeck2024 volume = {46}, year = {2024} } + +@article{Dorfler1996, + title={A convergent adaptive algorithm for Poisson’s equation}, + author={D{\"o}rfler, Willy}, + journal={SIAM Journal on Numerical Analysis}, + volume={33}, + number={3}, + pages={1106--1124}, + year={1996}, + publisher={SIAM} +} From 222a9cd20fe585e09b07ca03feef721111b91be0 Mon Sep 17 00:00:00 2001 From: Pablo Brubeck Date: Tue, 27 Jan 2026 16:09:03 +0000 Subject: [PATCH 6/7] add demo to TOC --- docs/source/advanced_tut.rst | 3 ++- tests/firedrake/demos/test_demos_run.py | 3 ++- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/docs/source/advanced_tut.rst b/docs/source/advanced_tut.rst index 084c605609..5007d12f4b 100644 --- a/docs/source/advanced_tut.rst +++ b/docs/source/advanced_tut.rst @@ -26,7 +26,7 @@ element systems. Full-waveform inversion: spatial and wave sources parallelism. 1D Vlasov-Poisson equation using vertical independent function spaces. Degree-independent multigrid convergence using patch relaxation. - Multigrid on adaptively-refined mesh hierarchies. + Multigrid on adaptively-refined mesh hierarchies. Monolithic multigrid with Vanka relaxation for Stokes. Vertex/edge star multigrid relaxation for H(div). Auxiliary space patch relaxation multigrid for H(curl). @@ -34,3 +34,4 @@ element systems. Shape optimisation. Steady Boussinesq problem with integral constraints. Steady multicomponent flow -- microfluidic mixing of hydrocarbons. + Deflation techniques for computing multiple solutions of nonlinear problems. diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py index 297f95a459..dd8ce626bc 100644 --- a/tests/firedrake/demos/test_demos_run.py +++ b/tests/firedrake/demos/test_demos_run.py @@ -18,11 +18,12 @@ DEMO_DIR = join(CWD, "..", "..", "..", "demos") SERIAL_DEMOS = [ - Demo(("adaptive_multigrid", "adaptive_multigrid"), ["netgen"]), + Demo(("adaptive_multigrid", "adaptive_multigrid"), ["netgen", "vtk"]), Demo(("benney_luke", "benney_luke"), ["vtk"]), Demo(("boussinesq", "boussinesq"), []), Demo(("burgers", "burgers"), ["vtk"]), Demo(("camassa-holm", "camassaholm"), ["vtk"]), + Demo(("deflation", "deflation"), ["matplotlib"]), Demo(("DG_advection", "DG_advection"), ["matplotlib"]), Demo(("eigenvalues_QG_basinmodes", "qgbasinmodes"), ["matplotlib", "slepc", "vtk"]), Demo(("extruded_continuity", 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