October 2015 - firedrake - imperial.ac.uk

Flux calculation
by Eike Mueller 03 Oct '15

03 Oct '15

Dear firedrakers if I have a scalar field phi and a vector field u, how can I calculate the flux f=phi*u in firedrake? I need to be able to loop over the dofs i of u, and for each one calculate f_i = sum_k phi_k*e_k(x_i)*u_i where e_k(x_i) is the k-th basis function of phi evaluated at the node x_i of u. But I somehow have to tell it that I want phi evaluated at u's nodes, not the other way round. Thanks, Eike Sent from my iPad

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Re: [firedrake] weirdness on bendy branch
by Fabio Luporini 02 Oct '15

02 Oct '15

Yes, what Lawrence says is correct. Hopefully the PR will be merged soon. Thanks. 2015-10-02 13:46 GMT+01:00 Mitchell, Lawrence < lawrence.mitchell(a)imperial.ac.uk>: > > > On 2 Oct 2015, at 13:10, Stephan Kramer <s.kramer(a)imperial.ac.uk> wrote: > > > > Hey guys > > > > I've been trying to implement a simple IP viscosity scheme for DG, but > was getting weird results, which I think I've tracked down to a bug related > to fd_bendy. > > > > If I run the following: > > > > from firedrake import * > > > > mesh2d = UnitSquareMesh(1,1) > > V = VectorFunctionSpace(mesh2d, "DG", 1) > > uv = Function(V) > > U_test = TestFunction(V) > > > > n = FacetNormal(mesh2d) > > F1 = dot(jump(U_test[0], n), jump(uv[0], n))*dS > > F2 = -dot(avg(grad(U_test[0])), jump(uv[0], n))*dS > > F=F1+F2 > > F0 = dot(jump(U_test[0],n)-avg(grad(U_test[0])), jump(uv[0], n))*dS > > > > M=assemble(derivative(F, uv)).M.values > > M1=assemble(derivative(F1, uv)).M.values > > M2=assemble(derivative(F2, uv)).M.values > > M0=assemble(derivative(F0, uv)).M.values > > Do you mind wrapping this up as a test (to merge against current master) > that ensures we don't regress when merging bendy? > > Lawrence > >

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Re: [firedrake] Solving this system of nonlinear equations
by David Ham 02 Oct '15

02 Oct '15

I'm not totally sure that's obvious. I agree that it's not the theoretically optimal solution, however it is well engineered which may make it a convenient solution, and if it's fast enough, then that's just fine. On Fri, 4 Sep 2015 at 11:22 Cotter, Colin J <colin.cotter(a)imperial.ac.uk> wrote: > PETSc shouldn't be used for this, as the problem is solved independently > at each node. > > --cjc > > On 4 September 2015 at 10:40, Ham, David A <david.ham(a)imperial.ac.uk> > wrote: > >> OK. I think I understand. In essence, assembling PETSc matrices is a core >> PyOP2 competence, so it *can* do this, although it'll be a little clunky >> because we don't have the full glory of Firedrake on top. >> >> I think it would help if you try to read some of the PyOP2 documentation >> for a bit of background: >> >> http://op2.github.io/PyOP2/concepts.html >> http://op2.github.io/PyOP2/kernels.html >> http://op2.github.io/PyOP2/linear_algebra.html >> http://op2.github.io/PyOP2/mixed.html >> >> The last one is important in this case because you're actually solving a >> mixed system over the mixed function space (A, B, C). >> >> Inconveniently, that means we need 3 kernels to assemble the residual and >> 7 kernels (one for each nonzero entry) to assemble the Jacobian. >> >> Once you have the code for assembling, I think it should be >> straightforward to provide PETSc4py with the right callbacks so that its >> variational inequality solver works. >> >> (Another possibility would be to bypass PyOP2 and write your own code >> accessing PETSc4py directly. That might actually be easier but assembly >> will be slower - which you may or may not care about). >> >> I don't have time right this instant to start writing out the Kernels (I >> have to teach on Monday and I'm not ready yet.) I'll try to get back to >> this later, or maybe someone else on the list will dive in and save me ;) >> >> Cheers, >> >> David >> > On Fri, 4 Sep 2015 at 09:37 Justin Chang <jychang48(a)gmail.com> wrote: >> > Hi everyone, >>> >>> Thanks for the input. So I was leaning more towards employing the newton >>> method for solving this. And yes, all the vectors live in the same function >>> space. If the system of equations were solved at each node i, the residual >>> algebraic vector of size 3x1 would be: >>> >>> r = [c_A(i)+c_C(i)-psi_A(i); >>> c_B(i)+c_C(i)-psi_B(i); >>> k1*c_A(i)*c_B(i)-c_C(i)] >>> >>> and the Jacobian 3x3 matrix would be: >>> >>> J = [1, 0, 1; >>> 0, 1, 1; >>> k1*c_B(i), k1*c_A(i), -1] >>> >>> Ideally I would like to have these expressed as a PETSc matrix and >>> vector because I would like to employ PETSc’s variational inequality solver >>> to enforce lower/upper bounds for c_A/B/C. >>> >>> As for the pyop2 option: >>> >>> void solve_system(double *c_A, double *c_B, double *c_C, double *k1, >>>>> double *psi_A, double *psi_B) >>>>> { >>>>> >>>>> // In here, you can dereference these variables to get the components >>>>> // of each variable at the node, e.g. >>>>> >>>>> // Set first component of c_A to sum of first two components of psi_A. >>>>> c_A[0] = psi_A[0] + psi_A[1]; >>>>> } >>>>> >>>>> I guess you can either write the newton (or fixed point) iteration out >>>>> by hand in the kernel (it shouldn't be too difficult), or else make library >>>>> calls (getting the correct link lines is possible I think). >>>>> >>>>> Now in your code you do: >>>>> >>>>> solve_kernel = op2.Kernel(kernel_code, "solve_system") >>>>> >>>>> op2.par_loop(solve_kernel, >>>>> c_A.dof_dset.node_set, >>>>> c_A.dat(op2.WRITE), >>>>> c_B.dat(op2.WRITE), >>>>> c_C.dat(op2.WRITE), >>>>> k1.dat(op2.READ), >>>>> psi_A.dat(op2.READ), >>>>> psi_B.dat(op2.READ)) >>>>> >>>>> >>>>> Does this make sense? >>>>> >>>> >>> I am guessing void solve_system(…) is a separate C file called >>> solve_system.c but how exactly do I get the components of each variable at >>> the node? So here’s how I did this with scipy: >>> >>> ================================ >>> >>> # Advection-diffusion firedrake code >>> # Compute psi_A, psi_B, and k1 ... >>> … >>> >>> from scipy.optimize import solve >>> import math >>> >>> # Solve system of nonlinear reactions >>> def equations(p,psi_A,psi_B,k_1): >>> c_A,c_B,c_C = p >>> return (c_A+c_C-psi_A, c_B+c_C-psi_B, c_C-k_1*c_A*c_B) >>> >>> # define vectors for primary species >>> c_A_vec = np.zeros(len(psi_A.vector())) >>> c_B_vec = np.zeros(len(psi_A.vector())) >>> >>> # define vector for secondary species >>> c_C_vec = np.zeros(len(psi_A.vector())) >>> >>> # Iterate through each node >>> for i in range(len(psi_A.vector())): >>> c_A_i,c_B_i,c_C_i = >>> fsolve(equations,(0.0,0.0,0.0),args=(psi_A.vector()[i],psi_B.vector()[i],k1) >>> c_A_vec[i] = c_A_i >>> c_B_vec[i] = c_B_i >>> c_C_vec[i] = c_C_i >>> >>> # Output >>> c_A = Function(Q) >>> c_B = Function(Q) >>> c_C = Function(Q) >>> c_A.vector()[:] = c_A_vec >>> c_B.vector()[:] = c_B_vec >>> c_C.vector()[:] = c_C_vec >>> File(“Solution_c_A.pvd”) << c_A >>> File(“Solution_c_B.pvd”) << c_B >>> File(“Solution_c_C.pvd”) << c_C >>> >>> =================================== >>> >>> The PyOP2 equivalent of the above would be a good starting point I >>> suppose. >>> >>> Thanks all, >>> Justin >>> >>> > -- > http://www.imperial.ac.uk/people/colin.cotter > > www.cambridge.org/9781107663916 > > >

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Re: [firedrake] The wonderful world of firedrake-install
by David Ham 02 Oct '15

02 Oct '15

Hi Justin, Sorry for the delay. I'm rushing backwards and forwards between the University of Reading and Imperial this week so lots of balls are getting dropped. The hdf5 problem is yet another "pip doesn't really look after dependencies" issue. I think we need to forcibly move the PETSc install earlier in the install process (because it's PETSc which installs the hdf5). I'll give that a try when I'm on a real internet connection. Your problem on your cluster is that the just in time compilation of one of your forms is failing. If you read the error log closely, it tells you which files failed to compile (in this case it's /scratch/306103/pyop2-cache-uid3003/c47dfce75aa72266c548fad137321af9.c) It also tells you where the related files containing the compile log and compile errors are. Could you post those files please? Cheers, David On Mon, 7 Sep 2015 at 00:01 Justin Chang <jychang48(a)gmail.com> wrote: > David, > > The error is still there even with the firedrake-install > from install-cython-workaround. For some strange reason I cannot link > the mip library to Cython using pip install ... but I compile Cython 0.23 > from source (./configure --prefix=/home/jchang23/.local && make && make > install) it links properly. I suspect this may have more to do with how my > local cluster's path dependencies behave. > > But now, i get this other error as shown in the attached > firedrake-install.log. I am guessing this has to do with not having an HDF5 > library available within my custom installed Python? > > Also, I note from inspection that apt-get normally installs the > blas/lapack libraries for you, so I added the tag '--download-fblaslapack' > to the petsc configure options. Is that the right thing to do or should I > somehow install blas/lapack as a whole so that all packages depend on it? > > Finally, I finally got firedrake to install properly when I loaded the > intel MPI compiled python libraries. They happened to have all of the most > up-to-date dependencies, and installed with 'python firedrake-install > --user --no_package_manager --disable_ssh'. When I run my programs on the > head node (and avoid getting yelled at), the RT0 code I have attached runs > fine, but when I submit a batch script as shown in runCompare, i get the > outputs as shown in output_error/test. Can you guys dissect what's going on > or would this be more of an issue that needs to be discussed with the > system admins? > > Thanks, > Justin > > > On Fri, Sep 4, 2015 at 4:14 AM, David Ham <David.Ham(a)imperial.ac.uk> > wrote: > >> Hi Justin, >> >> The prospective fix is in install-cython-workaround . Can you try that >> and tell me if it works for you? It will be merged to master as soon as >> Travis gives the green light. >> >> Cheers, >> >> David >> >> On Fri, 4 Sep 2015 at 09:42 Ham, David A <david.ham(a)imperial.ac.uk> >> wrote: >> >>> Hi Justin, >>> >>> Ah yes. I saw something like this when attempting to set up travis. The >>> problem appears to be that the dependency analysis in pip is broken so it >>> doesn't realise it has to build and install Cython before it builds h5py. I >>> think the workaround is to force Cython installation in the install script >>> before looking at the requirements file. I will push a branch presently and >>> merge it as soon as it passes tests. >>> >>> Cheers, >>> >>> David >>> >>> On Fri, 4 Sep 2015 at 08:17 Justin Chang <jychang48(a)gmail.com> wrote: >>> >>>> Hi David >>>> >>>> Thanks for the updated script. So I am using the system provided >>>> openmpi/1.8.3 module but built my own Python 2.7.10 and pip from source. I >>>> am getting an error as shown in the attached firedrake-install.log >>>> >>>> I also got a similar issue when I built my own openmpi library, >>>> although in that case I didn’t have the cython related errors and only had >>>> the ones related to hdf5/h5py >>>> >>>> Any idea what this is? >>>> >>>> Thanks, >>>> Justin >>>> >>>> >>>> On Sep 3, 2015, at 8:54 AM, David Ham <David.Ham(a)imperial.ac.uk> wrote: >>>> >>>> The branch is merged now so I believe this is now fixed. >>>> >>>> David >>>> >>>> On Thu, 3 Sep 2015 at 12:09 David Ham <David.Ham(a)imperial.ac.uk> wrote: >>>> >>>> It turns out that this is for the usual reason: I am an idiot. A fix >>>> for this is in the further-install-fixes branch and I will merge the branch >>>> as soon as tests pass. >>>> >>>> (What's actually going on is that I am abusing CalledProcessError when >>>> I should really raise a special exception for this case. I have now created >>>> a new exception and raised that instead). >>>> >>>> Cheers, >>>> >>>> David >>>> >>>> On Thu, 3 Sep 2015 at 01:56 Justin Chang <jychang48(a)gmail.com> wrote: >>>> >>>> Thanks David, >>>> >>>> I actually forgot about the --no_package_manager option. But when I >>>> tried installing with it, i get this error: >>>> >>>> $ python firedrake-install --no_package_manager >>>> Traceback (most recent call last): >>>> File "firedrake-install", line 265, in <module> >>>> raise subprocess.CalledProcessError >>>> TypeError: __init__() takes at least 3 arguments (1 given) >>>> >>>> Even if I add the --disable_ssh and --developer tags I get the above >>>> error. Am I suppose to do something else with this? >>>> >>>> Thanks, >>>> Justin >>>> >>>> On Tue, Sep 1, 2015 at 3:57 AM, David Ham <David.Ham(a)imperial.ac.uk> >>>> wrote: >>>> >>>> Thanks, >>>> >>>> On Sun, 30 Aug 2015 at 08:15 Justin Chang <jychang48(a)gmail.com> wrote: >>>> >>>> I agree that hype and a sparse direct solver like mumps is necessary. >>>> The latter needs --download-metis --download-parmetis >>>> --download-scalapack. >>>> >>>> >>>> Added and currently testing. >>>> >>>> >>>> More comments: >>>> >>>> 4) If mumps is to be included, the install script has to ensure that >>>> one has cmake >=2.5 when configuring metis, otherwise PETSc configure >>>> will return an error. Perhaps include "brew install cmake" and "sudo >>>> apt-get install cmake" in the script? >>>> >>>> >>>> Also added. Thanks for catching this. >>>> >>>> >>>> 5) Should --download-exodusii (and it's dependencies --download-netcdf >>>> --download-hdf5) also be included in the PETSC_CONFIGURE_OPTIONS? I >>>> imagine that if one wants to solve problems on large-scale >>>> unstructured grids, .exo files would be more feasible to use over .msh >>>> files. Or is exodusii already accounted for in firedrake/PyOP2/etc? >>>> >>>> >>>> And this. >>>> >>>> >>>> 6) If I run the script on a pristine MacOSX, I get an error saying >>>> "OSError: Unable to find SWIG installation. Please install SWIG >>>> version 2.0.0 or higher." I am guessing "brew install swig" was >>>> somehow missed in the firedrake-install script? >>>> >>>> Ah yes. Fecking swig. I've added the dependency to the new branch. >>>> Hopefully the swig dependency is going away very soon. >>>> >>>> >>>> >>>> 7) Perhaps not as important, but do y'all think it's possible to make >>>> this script more non-Ubuntu HPC system friendly? For isntance, either >>>> a) require the user to install his or her own local OpenMPI and Python >>>> libraries from source, b) let script download the tarballs/git >>>> repositories remotely and install it for you (kind of like how PETSc >>>> handles external packages through --download-<package>), or c) enable >>>> the user to point to the system provided OpenMPI and Python libraries. >>>> Same would need to be done for CMake, SWIG, and PCRE as these packages >>>> cannot simply be obtained from pip. >>>> >>>> >>>> I think (c) is the case right now. If you run the script on a >>>> non-Ubuntu system or pass the --no_package_manager option then it will just >>>> assume that you have installed the dependencies somewhere visible and will >>>> get on with pip. >>>> >>>> I think (b) is [a] hard and [b] in the case of MPI probably a bad idea. >>>> Supercomputers are such a diverse bunch of hacked up distributions that I >>>> think that installing the compiled dependencies automatically in a way >>>> which is portable would require a lot of install script work. In the case >>>> of MPI, you really need to use the MPI build which talks to the fast >>>> interconnect on your supercomputer, so automatically building vanilla >>>> OpenMPI is a Bad Thing. >>>> >>>> Cheers, >>>> >>>> David >>>> >>>> >>>> >>>> >>>> Thanks, >>>> Justin >>>> >>>> On Fri, Aug 28, 2015 at 7:20 AM, Lawrence Mitchell >>>> <lawrence.mitchell(a)imperial.ac.uk> wrote: >>>> > >>>> >> On 28 Aug 2015, at 14:01, David Ham <David.Ham(a)imperial.ac.uk> >>>> wrote: >>>> >> >>>> >> Hi All, >>>> >> >>>> >> I have a branch going through testing now which will handle >>>> PETSC_CONFIGURE_OPTIONS in a smarter way. Basically we'll just make sure >>>> that the things which are required are there, and we'll also honour >>>> anything the user has already set. I've also updated docs to say that >>>> that's what happens. >>>> >> >>>> >> I am also open to the suggestion that we could add more >>>> configuration options to the default set. Would anyone like to suggest what >>>> they should be? >>>> > >>>> > I think --download-hypre=1 at least. Maybe also a sparse direct >>>> solver (mumps?) which I think needs --download-metis --download-parmetis >>>> --download-mumps (maybe some others?) >>>> > >>>> > Lawrence >>>> > >>>> > >>>> > _______________________________________________ >>>> > 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 >>>> >>>> >>>> _______________________________________________ >>>> 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 >>>> >>>> >>>> >> _______________________________________________ >> firedrake mailing list >> firedrake(a)imperial.ac.uk >> https://mailman.ic.ac.uk/mailman/listinfo/firedrake >> >> >

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Re: [firedrake] JacobianDeterminant
by Andrew McRae 02 Oct '15

02 Oct '15

So you're using the 'bendy' branch in Firedrake as well? If so, JacobianDeterminant gets torn apart into components of grad(x) [x = coordinate field] within UFL. ufl/algorithms/change_to_reference.py: @memoized_handler def jacobian_determinant(self, o): domain = o.domain() J = self.jacobian(Jacobian(domain)) return determinant_expr(J) where determinant_expr is the obvious thing (use git grep determinant_expr from within a UFL subdirectory if you want to look at this) Note there's some magic that maps JacobianDeterminant to the handler jacobian_determinant. I think that pointing CellSize at JacobianDeterminant within Firedrake is something I'll revert in the future, since it's not backwards compatible for simplices (despite generalising well to other shapes). Will need to find another name, or just let people use the UFL object, like for FacetNormal. On 1 October 2015 at 21:01, Tuomas Karna <tuomas.karna(a)gmail.com> wrote: > Hi all, > > We're trying to figure out what CellSize actually does. > Does anyone know where JacobianDeterminant gets turn into code? > > - Tuomas > > > _______________________________________________ > firedrake mailing list > firedrake(a)imperial.ac.uk > https://mailman.ic.ac.uk/mailman/listinfo/firedrake >

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JacobianDeterminant
by Tuomas Karna 01 Oct '15

01 Oct '15

Hi all, We're trying to figure out what CellSize actually does. Does anyone know where JacobianDeterminant gets turn into code? - Tuomas

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Re: [firedrake] Expession
by Andrew McRae 01 Oct '15

01 Oct '15

Glad you found the problem :) Redefining the form can be a bad idea, since low-level C code will be generated for each new form (this is disastrous if you're in the "small mesh, lots of timesteps" regime!) A better way is to use Constant. For example, Q = Constant(0.0) a = Q*....*dx t = 0.0 while t < T - 0.5*dt: Q.assign(A*sin(2*pi*F*t)) ... t += dt On 1 October 2015 at 14:17, Anna Kalogirou <a.kalogirou(a)leeds.ac.uk> wrote: > I was just ready to send you a code when I realised what the problem was. > > When I define Q as a scalar expression, updating it after every time > step does not update the expression in the weak form (I define a and L > outside the time loop and then just call the solver at every time step). > Defining the expression by > > Q_expr = Expression("A*sin(2*pi*F*t)", t=t, A=A, F=F) > Q.interpolate(Q_expr) > > and updating it by Q_expr.t = t works though. > > Therefore I had to re-define the linear and bilinear parts of the weak > form inside the time loop for the first approach to work. > > > On 30/09/15 15:58, Lawrence Mitchell wrote: > > -----BEGIN PGP SIGNED MESSAGE----- > > Hash: SHA1 > > > > On 30/09/15 15:51, Anna Kalogirou wrote: > >> Yes, I double - and triple - checked and I believe I am doing the > >> same thing in both approaches. I don't understand then why the > >> results are different. > > Hmph, can you show us some breaking code? > > > > Lawrence > > > > -----BEGIN PGP SIGNATURE----- > > Version: GnuPG v1 > > > > iQEcBAEBAgAGBQJWC/imAAoJECOc1kQ8PEYvalEH/2z+lJMa3od/bKUbcbW73imx > > i3In1Wg9ZKVGeZG8VmVlcnirkr38EgzIi5ZoRZVtGQ6tmWsXTCwojfqsaLUv5k+b > > fWma1V9FWp8Vk6XSo8eqR+5oV0CBH0KRZtN8sZdGGqBm2QY5PBrQSyYcy5XHunIm > > h/mc7tT8oLRQKaPwYxuDt8WBtzc+BdNZPexLbGEa921qE1ms/SemDk85GbOQ5oqr > > oqvqfxpyrzwMBlPjz7GLN697u7r15e+qwGFni0Z6x3YsaBUF1OKXq7k0FvT5ObeI > > t30lYnhlxxKbbZQh1NI7goAZ7RUEd5RgAge2wUXN3V8EDyKVqzooRso1GGKGAwg= > > =jk88 > > -----END PGP SIGNATURE----- > > > > _______________________________________________ > > firedrake mailing list > > firedrake(a)imperial.ac.uk > > https://mailman.ic.ac.uk/mailman/listinfo/firedrake > > -- > > Dr Anna Kalogirou > Research Fellow > School of Mathematics > University of Leeds > > http://www1.maths.leeds.ac.uk/~matak/ > > > _______________________________________________ > firedrake mailing list > firedrake(a)imperial.ac.uk > https://mailman.ic.ac.uk/mailman/listinfo/firedrake >

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Re: [firedrake] Expession
by Andrew McRae 01 Oct '15

01 Oct '15

In the scalar approach, are you also redefining Q at each time step? ...just checking On 30 September 2015 at 15:46, Anna Kalogirou <a.kalogirou(a)leeds.ac.uk> wrote: > Yes I update the time component of the expession at every time step. > > > On 30/09/15 15:38, Andrew McRae wrote: > > Are you performing the interpolation every time step? > > On 30 September 2015 at 15:36, Anna Kalogirou <a.kalogirou(a)leeds.ac.uk> > wrote: > >> The solution behaves as expected when I define Q as a function, but in >> the other case there seems to be some kind of instability and I get a >> very strange result. >> >> Anna. >> >> >> On 30/09/15 15:28, Lawrence Mitchell wrote: >> > -----BEGIN PGP SIGNED MESSAGE----- >> > Hash: SHA1 >> > >> > On 30/09/15 15:19, Anna Kalogirou wrote: >> >> Dear all, >> >> >> >> I have a very simple, yet not so obvious - to me at least - >> >> question: >> >> >> >> If I define an expression which does not depend on space variables, >> >> for example >> >> >> >> Q_expr = Expression("A*sin(2*pi*F*t)", t=t, A=A, F=F) >> >> Q.interpolate(Q_expr) >> >> >> >> shouldn't this be equivalent to just writing Q as a scalar, >> >> >> >> Q = A*sin(2*pi*F*t) ? >> >> >> >> This quantity Q is then used in a weak form to multiply the test >> >> function v on a boundary, e.g. Q*v*ds_v(1). >> >> >> >> I get different results for the two cases. Clearly, in the 1st case >> >> Q is a function while in the 2nd case it is a scalar, but I don't >> >> see why this would give different results. I might be missing >> >> something obvious here... >> > These ought to be equivalent. But can you describe how they are >> > different? >> > >> > >> > Lawrence. >> > >> > -----BEGIN PGP SIGNATURE----- >> > Version: GnuPG v1 >> > >> > iQEcBAEBAgAGBQJWC/GPAAoJECOc1kQ8PEYvGncIAM7WI2JkrZ5Y1wZMyJAHNDJG >> > Ca476ZQcNCzKsZ/ntPL9SoXsbXu4Uu2mIVuT4r30VBc/NAEuBdkKppVmiwS3DTsP >> > qzLVP2nRL1ENxYWOUk8pWryE/jffyPP9ny5gHoRLPP/09vL3e4y8jKplw0bmKMgZ >> > UVvcp+kL1hqFujvCddxpym8gg4Y3UrMTdD2MaYJ8WHaFeqB9ZPNWomv24ZKKWhHU >> > HAms5sgsNedvdrgIY32AH2AARRysk4AsLsfjYymsiXPrOyPrlG5V2Muf4gKdqID1 >> > MRWiL7p6ZVxLnOytF4NtFRUItYLKcOrv487IIIHVfl+TVvmKO+IJpXDyx5wyZcY= >> > =EkO8 >> > -----END PGP SIGNATURE----- >> > >> > _______________________________________________ >> > firedrake mailing list >> > firedrake(a)imperial.ac.uk >> > https://mailman.ic.ac.uk/mailman/listinfo/firedrake >> >> -- >> >> Dr Anna Kalogirou >> Research Fellow >> School of Mathematics >> University of Leeds >> >> http://www1.maths.leeds.ac.uk/~matak/ >> >> >> _______________________________________________ >> 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 Anna Kalogirou > Research Fellow > School of Mathematics > University of Leeds > > http://www1.maths.leeds.ac.uk/~matak/ > >

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