# 2 Neumann Boundary Conditions **URL:** \ **Category:** dolfinx\ **Created:** [September 22, 2022, 8:45am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336 "2022-09-22T08:45:53Z")\ **Posts on this page:** 11\ **Page:** 1
**Author:** ![FenicsUser1](https://avatars.discourse-cdn.com/v4/letter/f/7bcc69/32.png) [@FenicsUser1](https://fenicsproject.discourse.group/u/FenicsUser1)\ **Post date:** [September 22, 2022, 8:45am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/1 "2022-09-22T08:45:53Z")
Hello how do I add 2 Neumann boundary conditions? My idea was: ```auto u1 = ufl.VersuchsFunktion(V) v1 = ufl.TestFunktion(V) dObs = Measure("ds", domain=mesh, subdomain_data=ft, subdomain_id=obstacle2_marker) dObt = Measure("ds", domain=mesh, subdomain_data=ft, subdomain_id=obstacle1_marker) a = ufl.inner(ufl.grad(u1), ufl.grad(v1)) * ufl.dx L = ufl.inner(E1,v1) * dObt + ufl.inner(E2, v1) * dObs u1 = fem.Function(V) problem1 = fem.petsc.LinearProblem(a, L, bcwall, petsc_options={"ksp_type": "preonly", "pc_type": "lu"}) p = problem1.solve() ``` but it does not work like this. ---
**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\ **Post date:** [September 22, 2022, 12:20pm UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/2 "2022-09-22T12:20:16Z")
See for instance [Setting multiple Dirichlet, Neumann, and Robin conditions — FEniCSx tutorial](https://jorgensd.github.io/dolfinx-tutorial/chapter3/robin_neumann_dirichlet.html) ---
**Author:** ![Marryweather](https://avatars.discourse-cdn.com/v4/letter/m/b782af/32.png) [@Marryweather](https://fenicsproject.discourse.group/u/Marryweather)\ **Post date:** [September 22, 2022, 6:21pm UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/3 "2022-09-22T18:21:22Z")
But there are not 2 Neumann boundary conditions executed at the same time, are they? ---
**Author:** ![FenicsUser1](https://avatars.discourse-cdn.com/v4/letter/f/7bcc69/32.png) [@FenicsUser1](https://fenicsproject.discourse.group/u/FenicsUser1)\ **Post date:** [September 22, 2022, 6:25pm UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/4 "2022-09-22T18:25:04Z")
I believe all 4 boundary conditions are executed, but I cannot yet understand how. Is there no easier way on my example? ---
**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\ **Post date:** [September 22, 2022, 7:48pm UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/5 "2022-09-22T19:48:31Z")
What your doing in your suggested example should work (up to some some spelling). If you can provide a minimal working example that does not do what you would expect people can help you. ---
**Author:** ![FenicsUser1](https://avatars.discourse-cdn.com/v4/letter/f/7bcc69/32.png) [@FenicsUser1](https://fenicsproject.discourse.group/u/FenicsUser1)\ **Post date:** [September 23, 2022, 7:58am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/6 "2022-09-23T07:58:23Z")
What do you mean by “up to some some spelling”? I am trying to make a minimal working example. ---
**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\ **Post date:** [September 23, 2022, 8:03am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/7 "2022-09-23T08:03:58Z")
> [@FenicsUser1](#): > > What do you mean by “up to some some spelling”? > [@FenicsUser1](#): > > ```auto > u1 = ufl.VersuchsFunktion(V) > v1 = ufl.TestFunktion(V) > > ``` These are German? names for the test and trial functon, and are thus not in the ufl namespace. ---
**Author:** ![funderal](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/funderal/32/6789_2.png) [@funderal](https://fenicsproject.discourse.group/u/funderal)\ **Post date:** [April 22, 2024, 9:23am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/8 "2024-04-22T09:23:56Z")
Hi I am trying to solve a problem with different materials. The code i am using is: ```auto from __future__ import print_function from dolfin import * from fenics import * # from boxfield import * import numpy as np import matplotlib.pyplot as plt def solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree=1, subdomains=[], linear_solver='Krylov', abs_tol=1E-5, rel_tol=1E-3, max_iter=1000): """ Solve -div(kappa*grad(u) = f on (0, 1) x (0, 1) with 2*Nx*Ny Lagrange elements of specified degree and u = u_D on the boundary. This version of the solver uses a specified combination of Dirichlet, Neumann, and Robin boundary conditions. """ # Create mesh and define function space mesh = UnitSquareMesh(Nx, Ny) V = FunctionSpace(mesh, 'P', degree) # Check if we have subdomains if subdomains: if not isinstance(kappa, (list, tuple, np.ndarray)): raise TypeError( 'kappa must be array if we have sudomains, not %s' % type(kappa)) materials = CellFunction('size_t', mesh) materials.set_all(0) for m, subdomain in enumerate(subdomains[1:], 1): subdomain.mark(materials, m) kappa_values = kappa V0 = FunctionSpace(mesh, 'DG', 0) kappa = Function(V0) help = np.asarray(materials.array(), dtype=np.int32) kappa.vector()[:] = np.choose(help, kappa_values) else: if not isinstance(kappa, (Expression, Constant)): raise TypeError( 'kappa is type %s, must be Expression or Constant' % type(kappa)) # Define boundary subdomains tol = 1e-14 class BoundaryX0(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[0], 0, tol) class BoundaryX1(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[0], 1, tol) class BoundaryY0(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[1], 0, tol) class BoundaryY1(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[1], 1, tol) # Mark boundaries boundary_markers = MeshFunction("size_t", mesh, mesh.topology().dim()-1, 0) boundary_markers.set_all(9999) bx0 = BoundaryX0() bx1 = BoundaryX1() by0 = BoundaryY0() by1 = BoundaryY1() bx0.mark(boundary_markers, 0) bx1.mark(boundary_markers, 1) by0.mark(boundary_markers, 2) by1.mark(boundary_markers, 3) # Redefine boundary integration measure ds = Measure('ds', domain=mesh, subdomain_data=boundary_markers) # Collect Dirichlet conditions bcs = [] for i in boundary_conditions: if 'Dirichlet' in boundary_conditions[i]: bc = DirichletBC(V, boundary_conditions[i]['Dirichlet'], boundary_markers, i) bcs.append(bc) debug1 = False if debug1: # Print all vertices that belong to the boundary parts for x in mesh.coordinates(): if bx0.inside(x, True): print('%s is on x = 0' % x) if bx1.inside(x, True): print('%s is on x = 1' % x) if by0.inside(x, True): print('%s is on y = 0' % x) if by1.inside(x, True): print('%s is on y = 1' % x) # Print the Dirichlet conditions print('Number of Dirichlet conditions:', len(bcs)) if V.ufl_element().degree() == 1: # P1 elements d2v = dof_to_vertex_map(V) coor = mesh.coordinates() for i, bc in enumerate(bcs): print('Dirichlet condition %d' % i) boundary_values = bc.get_boundary_values() for dof in boundary_values: print(' dof %2d: u = %g' % (dof, boundary_values[dof])) if V.ufl_element().degree() == 1: print(' at point %s' % (str(tuple(coor[d2v[dof]].tolist())))) # Define trial and test functions u = TrialFunction(V) v = TestFunction(V) # Collect Neumann integrals integrals_N = [] for i in boundary_conditions: if 'Neumann' in boundary_conditions[i]: if boundary_conditions[i]['Neumann'] != 0: g = boundary_conditions[i]['Neumann'] integrals_N.append(g*v*ds(i)) # Collect Robin integrals integrals_R_a = [] integrals_R_L = [] for i in boundary_conditions: if 'Robin' in boundary_conditions[i]: r, s = boundary_conditions[i]['Robin'] integrals_R_a.append(r*u*v*ds(i)) integrals_R_L.append(r*s*v*ds(i)) # Simpler Robin integrals integrals_R = [] for i in boundary_conditions: if 'Robin' in boundary_conditions[i]: r, s = boundary_conditions[i]['Robin'] integrals_R.append(r*(u - s)*v*ds(i)) # Sum integrals to define variational problem a = kappa*dot(grad(u), grad(v))*dx + sum(integrals_R_a) L = f*v*dx - sum(integrals_N) + sum(integrals_R_L) # Simpler variational problem F = kappa*dot(grad(u), grad(v))*dx + \ sum(integrals_R) - f*v*dx + sum(integrals_N) a, L = lhs(F), rhs(F) # Set linear solver parameters if linear_solver == 'Krylov': prm.linear_solver = 'gmres' prm.preconditioner = 'ilu' prm.krylov_solver.absolute_tolerance = abs_tol prm.krylov_solver.relative_tolerance = rel_tol prm.krylov_solver.maximum_iterations = max_iter else: prm.linear_solver = 'lu' # Compute solution u = Function(V) solve(a == L, u, bcs, solver_parameters=prm) return u def demo_bcs(): "Compute and plot solution using a combination of boundary conditions" # Define manufactured solution in sympy and derive f, g, etc. import sympy as sym x, y = sym.symbols('x[0], x[1]') # needed by UFL u = 1 + x **2 + 2*y** 2 # exact solution u_e = u # exact solution u_00 = u.subs(x, 0) # restrict to x = 0 u_01 = u.subs(x, 1) # restrict to x = 1 f = -sym.diff(u, x, 2) - sym.diff(u, y, 2) # -Laplace(u) f = sym.simplify(f) # simplify f g = -sym.diff(u, y).subs(y, 1) # compute g = -du/dn r = 1000 # Robin data, arbitrary s = u # Robin data, u = s # Collect variables variables = [u_e, u_00, u_01, f, g, r, s] # Turn into C/C++ code strings variables = [sym.printing.ccode(var) for var in variables] # Turn into FEniCS Expressions variables = [Expression(var, degree=2) for var in variables] # Extract variables u_e, u_00, u_01, f, g, r, s = variables # Define boundary conditions boundary_conditions = {0: {'Dirichlet': u_00}, # x = 0 1: {'Dirichlet': u_01}, # x = 1 2: {'Robin': (r, s)}, # y = 0 3: {'Neumann': g}} # y = 1 # Set linear solver parameters prm = LinearVariationalSolver.default_parameters() if linear_solver == 'Krylov': prm.linear_solver = 'gmres' prm.preconditioner = 'ilu' prm.krylov_solver.absolute_tolerance = abs_tol prm.krylov_solver.relative_tolerance = rel_tol prm.krylov_solver.maximum_iterations = max_iter else: prm.linear_solver = 'lu' # Compute solution kappa = Constant(1) Nx = Ny = 8 u = solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree=1, linear_solver='direct') # Compute maximum error at vertices mesh = u.function_space().mesh() vertex_values_u_e = u_e.compute_vertex_values(mesh) vertex_values_u = u.compute_vertex_values(mesh) error_max = np.max(np.abs(vertex_values_u_e - vertex_values_u)) print('error_max =', error_max) # Save and plot solution vtkfile = File('poisson_extended/solution_bcs.pvd') vtkfile << u plot(u) demo_bcs() ``` However, I always get this Error: ```auto NameError Traceback (most recent call last) Cell In[93], line 1 ----> 1 demo_bcs() Cell In[90], line 37, in demo_bcs() 35 # Set linear solver parameters 36 prm = LinearVariationalSolver.default_parameters() ---> 37 if linear_solver == 'Krylov': 38 prm.linear_solver = 'gmres' 39 prm.preconditioner = 'ilu' NameError: name 'linear_solver' is not defined ``` ---
**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\ **Post date:** [April 22, 2024, 5:01pm UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/9 "2024-04-22T17:01:54Z")
> [@funderal](#): > > ```auto > prm = LinearVariationalSolver.default_parameters() > if linear_solver == 'Krylov': > prm.linear_solver = 'gmres' > prm.preconditioner = 'ilu' > prm.krylov_solver.absolute_tolerance = abs_tol > prm.krylov_solver.relative_tolerance = rel_tol > prm.krylov_solver.maximum_iterations = max_iter > else: > prm.linear_solver = 'lu' > > ``` You never send `linear_solver` as an argument to `demo_bcs()`, and thus the variable does not exist. ---
**Author:** ![funderal](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/funderal/32/6789_2.png) [@funderal](https://fenicsproject.discourse.group/u/funderal)\ **Post date:** [April 23, 2024, 7:28am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/10 "2024-04-23T07:28:38Z")
Thank you for your answer, I am actually running the solver\_bcs and demo\_bcs functions in ft10\_possion\_extended and The code i am using is shown in the code below: ```auto from __future__ import print_function from fenics import * from boxfield import * import numpy as np def solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree=1, subdomains=[], linear_solver='Krylov', abs_tol=1E-5, rel_tol=1E-3, max_iter=1000): """ Solve -div(kappa*grad(u) = f on (0, 1) x (0, 1) with 2*Nx*Ny Lagrange elements of specified degree and u = u_D on the boundary. This version of the solver uses a specified combination of Dirichlet, Neumann, and Robin boundary conditions. """ # Create mesh and define function space mesh = UnitSquareMesh(Nx, Ny) V = FunctionSpace(mesh, 'P', degree) # Check if we have subdomains if subdomains: if not isinstance(kappa, (list, tuple, np.ndarray)): raise TypeError( 'kappa must be array if we have sudomains, not %s' % type(kappa)) materials = CellFunction('size_t', mesh) materials.set_all(0) for m, subdomain in enumerate(subdomains[1:], 1): subdomain.mark(materials, m) kappa_values = kappa V0 = FunctionSpace(mesh, 'DG', 0) kappa = Function(V0) help = np.asarray(materials.array(), dtype=np.int32) kappa.vector()[:] = np.choose(help, kappa_values) else: if not isinstance(kappa, (Expression, Constant)): raise TypeError( 'kappa is type %s, must be Expression or Constant' % type(kappa)) # Define boundary subdomains tol = 1e-14 class BoundaryX0(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[0], 0, tol) class BoundaryX1(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[0], 1, tol) class BoundaryY0(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[1], 0, tol) class BoundaryY1(SubDomain): def inside(self, x, on_boundary): return on_boundary and near(x[1], 1, tol) # Mark boundaries # boundary_markers = FacetFunction('size_t', mesh) boundary_markers = MeshFunction("size_t", mesh, mesh.topology().dim()-1, 0)######################修改 boundary_markers.set_all(9999) bx0 = BoundaryX0() bx1 = BoundaryX1() by0 = BoundaryY0() by1 = BoundaryY1() bx0.mark(boundary_markers, 0) bx1.mark(boundary_markers, 1) by0.mark(boundary_markers, 2) by1.mark(boundary_markers, 3) # Redefine boundary integration measure ds = Measure('ds', domain=mesh, subdomain_data=boundary_markers) # Collect Dirichlet conditions bcs = [] for i in boundary_conditions: if 'Dirichlet' in boundary_conditions[i]: bc = DirichletBC(V, boundary_conditions[i]['Dirichlet'], boundary_markers, i) bcs.append(bc) debug1 = False if debug1: # Print all vertices that belong to the boundary parts for x in mesh.coordinates(): if bx0.inside(x, True): print('%s is on x = 0' % x) if bx1.inside(x, True): print('%s is on x = 1' % x) if by0.inside(x, True): print('%s is on y = 0' % x) if by1.inside(x, True): print('%s is on y = 1' % x) # Print the Dirichlet conditions print('Number of Dirichlet conditions:', len(bcs)) if V.ufl_element().degree() == 1: # P1 elements d2v = dof_to_vertex_map(V) coor = mesh.coordinates() for i, bc in enumerate(bcs): print('Dirichlet condition %d' % i) boundary_values = bc.get_boundary_values() for dof in boundary_values: print(' dof %2d: u = %g' % (dof, boundary_values[dof])) if V.ufl_element().degree() == 1: print(' at point %s' % (str(tuple(coor[d2v[dof]].tolist())))) # Define trial and test functions u = TrialFunction(V) v = TestFunction(V) # Collect Neumann integrals integrals_N = [] for i in boundary_conditions: if 'Neumann' in boundary_conditions[i]: if boundary_conditions[i]['Neumann'] != 0: g = boundary_conditions[i]['Neumann'] integrals_N.append(g*v*ds(i)) # Collect Robin integrals integrals_R_a = [] integrals_R_L = [] for i in boundary_conditions: if 'Robin' in boundary_conditions[i]: r, s = boundary_conditions[i]['Robin'] integrals_R_a.append(r*u*v*ds(i)) integrals_R_L.append(r*s*v*ds(i)) # Simpler Robin integrals integrals_R = [] for i in boundary_conditions: if 'Robin' in boundary_conditions[i]: r, s = boundary_conditions[i]['Robin'] integrals_R.append(r*(u - s)*v*ds(i)) # Sum integrals to define variational problem a = kappa*dot(grad(u), grad(v))*dx + sum(integrals_R_a) L = f*v*dx - sum(integrals_N) + sum(integrals_R_L) # Simpler variational problem F = kappa*dot(grad(u), grad(v))*dx + \ sum(integrals_R) - f*v*dx + sum(integrals_N) a, L = lhs(F), rhs(F) # Set linear solver parameters if linear_solver == 'Krylov': prm.linear_solver = 'gmres' prm.preconditioner = 'ilu' prm.krylov_solver.absolute_tolerance = abs_tol prm.krylov_solver.relative_tolerance = rel_tol prm.krylov_solver.maximum_iterations = max_iter else: prm.linear_solver = 'lu' # Compute solution u = Function(V) solve(a == L, u, bcs, solver_parameters=prm) return u def demo_bcs(): "Compute and plot solution using a combination of boundary conditions" # Define manufactured solution in sympy and derive f, g, etc. import sympy as sym x, y = sym.symbols('x[0], x[1]') # needed by UFL u = 1 + x **2 + 2*y** 2 # exact solution u_e = u # exact solution u_00 = u.subs(x, 0) # restrict to x = 0 u_01 = u.subs(x, 1) # restrict to x = 1 f = -sym.diff(u, x, 2) - sym.diff(u, y, 2) # -Laplace(u) f = sym.simplify(f) # simplify f g = -sym.diff(u, y).subs(y, 1) # compute g = -du/dn r = 1000 # Robin data, arbitrary s = u # Robin data, u = s # Collect variables variables = [u_e, u_00, u_01, f, g, r, s] # Turn into C/C++ code strings variables = [sym.printing.ccode(var) for var in variables] # Turn into FEniCS Expressions variables = [Expression(var, degree=2) for var in variables] # Extract variables u_e, u_00, u_01, f, g, r, s = variables # Define boundary conditions boundary_conditions = {0: {'Dirichlet': u_00}, # x = 0 1: {'Dirichlet': u_01}, # x = 1 2: {'Robin': (r, s)}, # y = 0 3: {'Neumann': g}} # y = 1 # Compute solution kappa = Constant(1) Nx = Ny = 8 u = solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree=1, linear_solver='direct') # Compute maximum error at vertices mesh = u.function_space().mesh() vertex_values_u_e = u_e.compute_vertex_values(mesh) vertex_values_u = u.compute_vertex_values(mesh) error_max = np.max(np.abs(vertex_values_u_e - vertex_values_u)) print('error_max =', error_max) # Save and plot solution vtkfile = File('poisson_extended/solution_bcs.pvd') vtkfile << u plot(u) demo_bcs() ``` The first problem I am having is shown below: ```auto ModuleNotFoundError Traceback (most recent call last) Cell In[52], line 4 1 from __future__ import print_function 3 from fenics import * ----> 4 from boxfield import * 5 import numpy as np 7 def solver_bcs(kappa, f, boundary_conditions, Nx, Ny, 8 degree=1, 9 subdomains=[], (...) 12 rel_tol=1E-3, 13 max_iter=1000): ModuleNotFoundError: No module named 'boxfield' ``` ```auto NameError Traceback (most recent call last) Cell In[51], line 1 ----> 1 demo_bcs() Cell In[50], line 38, in demo_bcs() 36 kappa = Constant(1) 37 Nx = Ny = 8 ---> 38 u = solver_bcs(kappa, f, boundary_conditions, Nx, Ny, 39 degree=1, linear_solver='direct') 41 # Compute maximum error at vertices 42 mesh = u.function_space().mesh() Cell In[49], line 162, in solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree, subdomains, linear_solver, abs_tol, rel_tol, max_iter) 160 prm.krylov_solver.maximum_iterations = max_iter 161 else: --> 162 prm.linear_solver = 'lu' 164 # Compute solution 165 u = Function(V) NameError: name 'prm' is not defined ``` ---
**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\ **Post date:** [April 23, 2024, 8:44am UTC](https://fenicsproject.discourse.group/t/2-neumann-boundary-conditions/9336/11 "2024-04-23T08:44:00Z")
> [@funderal](#): > > ```auto > ModuleNotFoundError Traceback (most recent call last) > Cell In[52], line 4 > 1 from __future__ import print_function > 3 from fenics import * > ----> 4 from boxfield import * > 5 import numpy as np > 7 def solver_bcs(kappa, f, boundary_conditions, Nx, Ny, > 8 degree=1, > 9 subdomains=[], > (...) > 12 rel_tol=1E-3, > 13 max_iter=1000): > > ModuleNotFoundError: No module named 'boxfield' > > ``` You need to install the boxfield package. Please note that boxfield and legacy FEniCS is deprecated. > [@funderal](#): > > ```auto > NameError Traceback (most recent call last) > Cell In[51], line 1 > ----> 1 demo_bcs() > > Cell In[50], line 38, in demo_bcs() > 36 kappa = Constant(1) > 37 Nx = Ny = 8 > ---> 38 u = solver_bcs(kappa, f, boundary_conditions, Nx, Ny, > 39 degree=1, linear_solver='direct') > 41 # Compute maximum error at vertices > 42 mesh = u.function_space().mesh() > > Cell In[49], line 162, in solver_bcs(kappa, f, boundary_conditions, Nx, Ny, degree, subdomains, linear_solver, abs_tol, rel_tol, max_iter) > 160 prm.krylov_solver.maximum_iterations = max_iter > 161 else: > --> 162 prm.linear_solver = 'lu' > 164 # Compute solution > 165 u = Function(V) > > NameError: name 'prm' is not defined > > ``` Similar error as the first one, you have not defined `prm` inside `solver_bcs`. Please look carefully through your code before posting, as the error message literally tells you what is wrong.