- firedrake - imperial.ac.uk

error norm convergence issues for RT/BDM
by Justin Chang 01 Jun '17

01 Jun '17

Hi all, I am playing around with the RT/BDM L2 error norm convergence for the Darcy equation: mu/K*v + grad(p) = rhob div(v) = 0 Where mu is viscosity, K is permeability tensor, and rhob is specific body force. The analytical solutions for a bi-unit square are: p(x,y) = sin(pi*x)*sin(pi*y) v_x(x,y) = sin(pi*x)*cos(pi*x) v_y(x,y) = -cos(pi*x)*sin(pi*y) And mu = K = 1. Theory states that RT0 has velocity/pressure slope of 1.0 and that BDM has velocity slope of 2.0. I get these or better. Now if I extend the Darcy formulation to include pressure-dependence for viscosity: mu(p)/K*v + grad(p) = rhob div(v) = f where mu(p) = mu*(1 + beta*p) So we now have a nonlinear formulation. By employing the same analytical solutions, the only thing that changes is rhob, but the velocity L2 errornorm slopes for both RT and BDM worsen. For example, BDM goes from 2.0 to 1.0. Pressure slopes remains unchanged. Why is this happening? Does this have something to do with the DG0 pressure functionspace that is now included in the mass matrix term? If I used a stabilized equal-order formulation for velocity/pressure, no such slope differences were observed. These trends were also observed with FEniCS. Attached is the code y'all can play around with: Run it as follows: python H_conv_barus.py Darcy RT 10 python H_conv_barus.py Darcy RT 20 python H_conv_barus.py Darcy RT 40 ... python H_conv_barus.py Darcy BDM 10 python H_conv_barus.py Darcy BDM 20 python H_conv_barus.py Darcy BDM 40 ... python H_conv_barus.py Barus RT 10 python H_conv_barus.py Barus RT 20 python H_conv_barus.py Barus RT 40 ... python H_conv_barus.py Barus BDM 10 python H_conv_barus.py Barus BDM 20 python H_conv_barus.py Barus BDM 40 ... I am trying to understand whether these H(div) elements are suitable for nonlinear mixed formulations or if there is a better way of implementing what I have. Thanks! Justin

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Different rates of convergence in parallel and serial using BDM and RT
by Sander Rhebergen 25 May '17

25 May '17

Hello, I'm quite new to using Firedrake (although I have used FEniCS in the past). Based on some of the Firedrake demos I wrote a Stokes code in which I'm using a mixed BDM/DG or RT/DG function space (see attached). In this code I compare my numerical solution to an exact solution on different grids and computed the rates of convergence. This runs fine in serial but in parallel the rates of convergence are not as I expect. For example, in serial, rates of convergence using BDM(order 2)/DG (order 1) are: Err_u r Err_p r div(u) r 8.21e-05 2.97 9.98e-03 1.95 5.17e-09 -0.80 1.04e-05 2.98 2.54e-03 1.97 7.75e-09 -0.59 1.31e-06 2.99 6.41e-04 1.99 5.43e-09 0.51 but in parallel (mpiexec -np 4 python stokes_hdiv.py) Err_u r Err_p r div(u) r 8.18e-05 3.48 9.99e-03 2.07 2.99e-09 0.22 5.07e-05 0.69 2.91e-03 1.78 5.69e-09 -0.93 3.48e-05 0.54 1.34e-03 1.12 7.88e-09 -0.47 Am I missing something in my code to deal with runs in parallel? Using a standard Taylor-Hood P2-P1 finite element (code is also attached), I am getting the correct rates of convergence for this Stokes problem. Thanks, Sander

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Function at() PointNotInDomainError
by Pan, Wei 24 May '17

24 May '17

Hey guys, I'm having problems with the Function.at() function. Below is an example code from __future__ import division from firedrake import * import numpy as np mesh_base = SquareMesh(8, 8, 1.) mesh_hierarchy = MeshHierarchy(mesh_base, 6) fine_msh = mesh_hierarchy[6] # 512 x 512 mesh fine_dg_space = FunctionSpace(fine_msh, "DG", 1) x, y = SpatialCoordinate(fine_dg_space) q0 = Function(fine_dg_space).interpolate(x) x_array = np.array([_x / 512.0 for _x in range(513)]) # coordinates # x_array[0] += 1. / 1000000.0 # a slight nudge away from boundary also fails for i in x_array: for j in x_array: print [i, j], q0.at(np.array([i, j])) My domain is [0, 1], mesh size 512x512. I keep on getting the following PointNotInDomainError when I run the above, where 0.43164062 correspond to 221 of 512. What am I doing wrong? [0.0, 0.427734375] 0.0 [0.0, 0.4296875] 0.0 [0.0, 0.431640625] Traceback (most recent call last): File "/home/wpan1/PycharmProjects/MyFirstProject/test_at.py", line 22, in <module> print [i, j], q0.at(np.array([i, j])) File "/home/wpan1/Documents/firedrake/src/firedrake/firedrake/function.py", line 574, in at raise PointNotInDomainError(self.function_space().mesh(), points[i].reshape(-1)) firedrake.function.PointNotInDomainError: domain <Mesh #7> does not contain point [ 0. 0.43164062] Thanks Wei

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Failing tests on subprojects of Firedrake
by Gregory, Alastair C A 24 May '17

24 May '17

Hi Firedrakers, I'm having some problems passing any tests (even master fails) on my 2 subprojects of firedrake: firedrake-mlmc and flooddrake. All automated tests and linting tests are absolutely fine on my local machine with a version of Firedrake updated today. If possible, could someone please look at these and see if anything is obviously wrong on the Travis testing side of these two subprojects (i.e. with the travis script etc.)? Many Thanks, Alastair Gregory Imperial College London Office 749 Huxley Building, South Kensington (Tel: 07794 243913) | (Email: a.gregory14(a)imperial.ac.uk)

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Re: [firedrake] imposing Dirichlet BCS
by David Ham 16 May '17

16 May '17

Hi Francis, Usually we use the boundary markers on the incoming mesh. Where are you getting your more general mesh from? Regards, David On Wed, 10 May 2017 at 15:37 Francis Poulin <fpoulin(a)uwaterloo.ca> wrote: > Hello, > > We are trying to translate part of the mixed-poisson Fenics demo into > Firedrake. I know that there is already a Firedrake demo that does this > but for the special case of a square. I am hoping we can do this more > generally. > > Below are the lines of code from Fenics. Can someone suggest how this > could be implemented in Firedrake? > > Cheers, Francis > > > https://fenics.readthedocs.io/projects/dolfin/en/latest/demos/mixed-poisson… > > # Define function G such that G \cdot n = gclass BoundarySource(Expression): > def __init__(self, mesh, **kwargs): > self.mesh = mesh > def eval_cell(self, values, x, ufc_cell): > cell = Cell(self.mesh, ufc_cell.index) > n = cell.normal(ufc_cell.local_facet) > g = sin(5*x[0]) > values[0] = g*n[0] > values[1] = g*n[1] > def value_shape(self): > return (2,) > G = BoundarySource(mesh, degree=2) > # Define essential boundarydef boundary(x): > return x[1] < DOLFIN_EPS or x[1] > 1.0 - DOLFIN_EPS > > > bc = DirichletBC(W.sub(0), G, boundary) > > ------------------ > Francis Poulin > Associate Professor > Department of Applied Mathematics > University of Waterloo > > email: fpoulin(a)uwaterloo.ca > Web: https://uwaterloo.ca/poulin-research-group/ > Telephone: +1 519 888 4567 x32637 <(519)%20888-4567> > > -- Dr David Ham Department of Mathematics Imperial College London

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Re: [firedrake] Solver error -- Provided residual is not a linear form
by David Ham 15 May '17

15 May '17

Actually why extract the values and then re-interpolate? Can't you just set x=mesh.coordinate and use x[0] in the expression? On Mon, 15 May 2017 at 13:59 Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> wrote: > Hi David, > > Thank you for spotting my mistake. Indeed, I should have interpolated x[0] > into a Function. > > Regards, > > Anna. > > On 15 May 2017, at 13:49, David Ham <David.Ham(a)imperial.ac.uk> wrote: > > Hi Anna, > > The proximate cause of your error is line 24 of solvers.py which contains > multiple subexpressions of the form: > > Z0 - H - tan(a)*(x-L)) > > Here, Z0 is a Function but x is a numpy array containing coordinate > values. You are therefore subtracting a numpy array from a Function. This > makes no mathematical sense and, unsurprisingly, causes UFL to choke. > > Regards, > > David > > > > On Mon, 15 May 2017 at 13:25 Homolya, Miklós <m.homolya14(a)imperial.ac.uk> > wrote: > >> If you want to preserve the symbolic structure of maximum and its >> dependence on u, then you need neither interpolate nor assign. >> >> maximum = 0.5*(u + abs(u)) >> F = v*(u + maximum)*dx >> >> >> On 15/05/17 10:06, Anna Kalogirou wrote: >> >> Hi David, >> >> What I would like to do is to define a residual that includes some >> nonlinear function depending on the unknown, similar to the example with >> the maximum below (ultimately, I need to use functions abs, exp, tan). I do >> want to dependence of maximum on u to appear in the definition of the >> residual somehow; how can I do that then? >> >> Thanks, >> >> Anna. >> >> On 15 May 2017, at 09:44, David Ham <David.Ham(a)imperial.ac.uk> wrote: >> >> >> >> On Sun, 14 May 2017 at 21:31 Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> >> wrote: >> >>> Following up on my previous email, I have found a workaround to those >>> issues, but now I have a problem with the solvers: >>> >>> Traceback (most recent call last): >>> File >>> "/Users/matak/Downloads/Inequality-constraint-Rockafellar/solvers.py", line >>> 32, in solver_F >>> F_problem = NonlinearVariationalProblem(F_solve, w)#, aP=Ap) >>> File >>> "/Users/matak/firedrake/src/firedrake/firedrake/variational_solver.py", >>> line 47, in __init__ >>> raise ValueError("Provided residual is not a linear form") >>> ValueError: Provided residual is not a linear form >>> >>> There two issues here: >>> 1. I cannot give a preconditioning matrix aP -- is this not possible for >>> nonlinear problems? >>> >> >> As the documentation for NonlinearVariationalProblem suggests, you can >> provide Jp, a UFL form for the preconditioning Jacobian. This is the >> equivalent of aP. >> >> >>> 2. I suspect the error above is because I have functions that depend on >>> the solution, and I don’t know if that is acceptable in firedrake. I have >>> something like the following: >>> >>> u = Function(V) >>> v = TestFunction(V) >>> maximum = interpolate(0.5*(u + abs(u)), V) >>> F = v*(u + maximum)*dx >>> >> >> Actually, as Colin sort of says, as far as Firedrake is concerned, >> maximum does not depend on u. This is because doing the interpolation >> breaks the symbolic link between u and maximum. In the code you have >> written, maximum does not get updated inside the variational solver as the >> guess for u gets updated. Whether that is what you wanted to do or not >> depends on the maths of your problem. >> >> The error you are seeing, however, is caused by the Form having two >> arguments. This almost certainly means you have a TrialFunction in your >> nonlinear problem, which makes no sense. >> >> Regards, >> >> David >> >> >>> >>> Best wishes, >>> >>> Anna. >>> >> -- >> Dr David Ham >> Department of Mathematics >> Imperial College London >> >> _______________________________________________ >> firedrake mailing list >> firedrake(a)imperial.ac.uk >> https://mailman.ic.ac.uk/mailman/listinfo/firedrake >> >> >> >> _______________________________________________ >> firedrake mailing listfiredrake@imperial.ac.ukhttps://mailman.ic.ac.uk/mailman/listinfo/firedrake >> >> -- > Dr David Ham > Department of Mathematics > Imperial College London > _______________________________________________ > firedrake mailing list > firedrake(a)imperial.ac.uk > https://mailman.ic.ac.uk/mailman/listinfo/firedrake > > > -- Dr David Ham Department of Mathematics Imperial College London

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Re: [firedrake] Solver error -- Provided residual is not a linear form
by David Ham 15 May '17

15 May '17

Hi Anna, The proximate cause of your error is line 24 of solvers.py which contains multiple subexpressions of the form: Z0 - H - tan(a)*(x-L)) Here, Z0 is a Function but x is a numpy array containing coordinate values. You are therefore subtracting a numpy array from a Function. This makes no mathematical sense and, unsurprisingly, causes UFL to choke. Regards, David On Mon, 15 May 2017 at 13:25 Homolya, Miklós <m.homolya14(a)imperial.ac.uk> wrote: > If you want to preserve the symbolic structure of maximum and its > dependence on u, then you need neither interpolate nor assign. > > maximum = 0.5*(u + abs(u)) > F = v*(u + maximum)*dx > > > On 15/05/17 10:06, Anna Kalogirou wrote: > > Hi David, > > What I would like to do is to define a residual that includes some > nonlinear function depending on the unknown, similar to the example with > the maximum below (ultimately, I need to use functions abs, exp, tan). I do > want to dependence of maximum on u to appear in the definition of the > residual somehow; how can I do that then? > > Thanks, > > Anna. > > On 15 May 2017, at 09:44, David Ham <David.Ham(a)imperial.ac.uk> wrote: > > > > On Sun, 14 May 2017 at 21:31 Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> > wrote: > >> Following up on my previous email, I have found a workaround to those >> issues, but now I have a problem with the solvers: >> >> Traceback (most recent call last): >> File >> "/Users/matak/Downloads/Inequality-constraint-Rockafellar/solvers.py", line >> 32, in solver_F >> F_problem = NonlinearVariationalProblem(F_solve, w)#, aP=Ap) >> File >> "/Users/matak/firedrake/src/firedrake/firedrake/variational_solver.py", >> line 47, in __init__ >> raise ValueError("Provided residual is not a linear form") >> ValueError: Provided residual is not a linear form >> >> There two issues here: >> 1. I cannot give a preconditioning matrix aP -- is this not possible for >> nonlinear problems? >> > > As the documentation for NonlinearVariationalProblem suggests, you can > provide Jp, a UFL form for the preconditioning Jacobian. This is the > equivalent of aP. > > >> 2. I suspect the error above is because I have functions that depend on >> the solution, and I don’t know if that is acceptable in firedrake. I have >> something like the following: >> >> u = Function(V) >> v = TestFunction(V) >> maximum = interpolate(0.5*(u + abs(u)), V) >> F = v*(u + maximum)*dx >> > > Actually, as Colin sort of says, as far as Firedrake is concerned, maximum > does not depend on u. This is because doing the interpolation breaks the > symbolic link between u and maximum. In the code you have written, maximum > does not get updated inside the variational solver as the guess for u gets > updated. Whether that is what you wanted to do or not depends on the maths > of your problem. > > The error you are seeing, however, is caused by the Form having two > arguments. This almost certainly means you have a TrialFunction in your > nonlinear problem, which makes no sense. > > Regards, > > David > > >> >> Best wishes, >> >> Anna. >> > -- > Dr David Ham > Department of Mathematics > Imperial College London > > _______________________________________________ > firedrake mailing list > firedrake(a)imperial.ac.uk > https://mailman.ic.ac.uk/mailman/listinfo/firedrake > > > > _______________________________________________ > firedrake mailing listfiredrake@imperial.ac.ukhttps://mailman.ic.ac.uk/mailman/listinfo/firedrake > > -- Dr David Ham Department of Mathematics Imperial College London

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Present installation issues with PETSc build
by Lawrence Mitchell 15 May '17

15 May '17

Dear firedrakers, some of you attempting to install or update recently may have seen that the PETSc build fails (attempting to install Eigen). This is a reported bug in the PETSc installer, we hope that it will be fixed in the next day or so. As a short-term workaround, you can disable the installation of Eigen when running firedrake-install or firedrake-update like so: $ PETSC_CONFIGURE_OPTIONS="--download-eigen=0" firedrake-install or $ PETSC_CONFIGURE_OPTIONS="--download-eigen=0" firedrake-update This will disable the ability to solve problems using hybridisation methods, but Firedrake will otherwise be functional. A subsequent call to firedrake-install or firedrake-update without setting PETSC_CONFIGURE_OPTIONS will subsequently download and install Eigen, you can do this when the problem is fixed. I will update this thread when the bug fix arrives in PETSc and we have confirmed it works for us. Thanks, and apologies for any problems this might be causing, Lawrence

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Re: [firedrake] Solver error -- Provided residual is not a linear form
by David Ham 15 May '17

15 May '17

On Sun, 14 May 2017 at 21:31 Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> wrote: > Following up on my previous email, I have found a workaround to those > issues, but now I have a problem with the solvers: > > Traceback (most recent call last): > File > "/Users/matak/Downloads/Inequality-constraint-Rockafellar/solvers.py", line > 32, in solver_F > F_problem = NonlinearVariationalProblem(F_solve, w)#, aP=Ap) > File > "/Users/matak/firedrake/src/firedrake/firedrake/variational_solver.py", > line 47, in __init__ > raise ValueError("Provided residual is not a linear form") > ValueError: Provided residual is not a linear form > > There two issues here: > 1. I cannot give a preconditioning matrix aP -- is this not possible for > nonlinear problems? > As the documentation for NonlinearVariationalProblem suggests, you can provide Jp, a UFL form for the preconditioning Jacobian. This is the equivalent of aP. > 2. I suspect the error above is because I have functions that depend on > the solution, and I don’t know if that is acceptable in firedrake. I have > something like the following: > > u = Function(V) > v = TestFunction(V) > maximum = interpolate(0.5*(u + abs(u)), V) > F = v*(u + maximum)*dx > Actually, as Colin sort of says, as far as Firedrake is concerned, maximum does not depend on u. This is because doing the interpolation breaks the symbolic link between u and maximum. In the code you have written, maximum does not get updated inside the variational solver as the guess for u gets updated. Whether that is what you wanted to do or not depends on the maths of your problem. The error you are seeing, however, is caused by the Form having two arguments. This almost certainly means you have a TrialFunction in your nonlinear problem, which makes no sense. Regards, David > > Best wishes, > > Anna. > -- Dr David Ham Department of Mathematics Imperial College London

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Re: [firedrake] Solver error -- Provided residual is not a linear form
by Andrew McRae 15 May '17

15 May '17

Okay, I'm not sure what's going on here, so I'll leave this for someone else. I suspect that one of those variables is "not what you think it is", i.e. not a UFL object. Perhaps you accidentally overwrote a variable, or just set it to the wrong object? The original error message turns out to be misleading. Apparently the as_ufl call in ufl/constantvalue.py is failing, which is the real error. In trying to generate an error message for that, it tries to generate a string representation of the expression, which also failed, producing the error that you reported. Full traceback: (firedrake) andrew@ubuntu:~/anna/buoy2d/Inequality-constraint-Rockafellar$ python buoy-swe.py UFL:ERROR UFL conditions cannot be evaluated as bool in a Python context. Traceback (most recent call last): File "buoy-swe.py", line 82, in <module> F_solver = solver_F(phi0_5, h1, lambda0_5, Z1, W0_5, I, w, phi0, h0, Z0, W0, phi_t, h_t, lambda_t, Z_t, W_t, I_t, v1, v2, v3, v4, v5, v6, dt, Hx, H0, L, dR_dt, g, rho, Mass, H, a, r, coords.dat.data, solvers_print); File "/home/andrew/anna/buoy2d/Inequality-constraint-Rockafellar/solvers.py", line 25, in solver_F + 0.5*((lambda0_5 + exp(r*(h0-(Z0 - H - tan(a)*(x-L)))) - 1) + abs(lambda0_5 + exp(r*(h0-(Z0 - H - tan(a)*(x-L)))) - 1))*exp(r*(h0-(Z0 - H - tan(a)*(x-L)))) )*dx File "/home/andrew/firedrake/src/ufl/ufl/operators.py", line 584, in exp return _mathfunction(f, Exp) File "/home/andrew/firedrake/src/ufl/ufl/operators.py", line 570, in _mathfunction f = as_ufl(f) File "/home/andrew/firedrake/src/ufl/ufl/constantvalue.py", line 417, in as_ufl " to any UFL type." % str(expression)) File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/numeric.py", line 1961, in array_str return array2string(a, max_line_width, precision, suppress_small, ' ', "", str) File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/arrayprint.py", line 463, in array2string separator, prefix, formatter=formatter) File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/arrayprint.py", line 327, in _array2string suppress_small, formatter) File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/arrayprint.py", line 290, in _get_format_function formatdict = _get_formatdict(data, precision, suppress_small, formatter) File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/arrayprint.py", line 239, in _get_formatdict 'int': IntegerFormat(data), File "/home/andrew/firedrake/local/lib/python2.7/site-packages/numpy/core/arrayprint.py", line 654, in __init__ max_str_len = max(len(str(maximum.reduce(data))), File "/home/andrew/firedrake/src/ufl/ufl/conditional.py", line 46, in __bool__ error("UFL conditions cannot be evaluated as bool in a Python context.") File "/home/andrew/firedrake/src/ufl/ufl/log.py", line 172, in error raise self._exception_type(self._format_raw(*message)) ufl.log.UFLException: UFL conditions cannot be evaluated as bool in a Python context. On 15 May 2017 at 10:35, Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> wrote: > The full code can be found here: > https://bitbucket.org/annakalog/buoy2d/src/7a99dc59b9d1ddd90105d1313c83ee > a71baafda4/Inequality-constraint-Rockafellar/?at=master > Please make sure to go to the right folder, i.e. > Buoy2D <https://bitbucket.org/annakalog/buoy2d/src/7a99dc59b9d1?at=master> > / Inequality-constraint-Rockafellar / > > Thanks, > > Anna. > > On 15 May 2017, at 10:13, Andrew McRae <A.T.T.McRae(a)bath.ac.uk> wrote: > > Unless I'm being dumb, this really should work as you've written it (using > assign/interpolate definitely *won't* do what you want). > > Can you send a fuller version of your code as I suspect the error is > caused by something else? > > Andrew > > On 15 May 2017 at 09:48, Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> wrote: > >> Hi Colin, >> >> Thanks, indeed I had forgotten a trial function in my residual. >> >> Yes I do need the the u-dependence in the maximum during the solve. If I >> substitute the definition of maximum into the residual, i.e. write >> F = v*(u + 0.5*(u + abs(u)))*dx >> gives the error >> ufl.log.UFLException: UFL conditions cannot be evaluated as bool in a >> Python context. >> >> Will it work if I use assign instead of interpolate? >> >> Best, Anna. >> >> On 15 May 2017, at 07:59, Colin Cotter <colin.cotter(a)imperial.ac.uk> >> wrote: >> >> Hi Anna, >> Are you intending that the u-dependence in maximum should be maintained >> during the solve? After doing the interpolation, there will be no symbolic >> record of where maximum's values came from. >> >> That's probably not the cause of the error message though. It says that >> the form is not a linear form. This might mean that you accidentally >> included some TrialFunction (which turns it into a bilinear form) or >> perhaps forgot a factor of dx or dS somewhere (which stops it from being a >> form). >> >> all the best >> --cjc >> >> On 14 May 2017 at 21:30, Anna Kalogirou <A.Kalogirou(a)leeds.ac.uk> wrote: >> >>> Following up on my previous email, I have found a workaround to those >>> issues, but now I have a problem with the solvers: >>> >>> Traceback (most recent call last): >>> File "/Users/matak/Downloads/Inequality-constraint-Rockafellar/solvers.py", >>> line 32, in solver_F >>> F_problem = NonlinearVariationalProblem(F_solve, w)#, aP=Ap) >>> File "/Users/matak/firedrake/src/firedrake/firedrake/variational_solver.py", >>> line 47, in __init__ >>> raise ValueError("Provided residual is not a linear form") >>> ValueError: Provided residual is not a linear form >>> >>> There two issues here: >>> 1. I cannot give a preconditioning matrix aP -- is this not possible for >>> nonlinear problems? >>> 2. I suspect the error above is because I have functions that depend on >>> the solution, and I don’t know if that is acceptable in firedrake. I have >>> something like the following: >>> >>> u = Function(V) >>> v = TestFunction(V) >>> maximum = interpolate(0.5*(u + abs(u)), V) >>> F = v*(u + maximum)*dx >>> >>> Best wishes, >>> >>> Anna. >>> >> >> >> >> -- >> http://www.imperial.ac.uk/people/colin.cotter >> >> www.cambridge.org/9781107663916 >> >> >> _______________________________________________ >> firedrake mailing list >> firedrake(a)imperial.ac.uk >> https://mailman.ic.ac.uk/mailman/listinfo/firedrake >> >> >> > _______________________________________________ > firedrake mailing list > firedrake(a)imperial.ac.uk > https://mailman.ic.ac.uk/mailman/listinfo/firedrake > > >

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