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Introduction
Variational Formulation
Commented Program
Results
Complete Source Code

Introduction

In this tutorial the implementation of inhomogeneous Dirichlet boundary conditions using a Dirichlet lift ansatz is shown.

The equation solved is the reaction-diffusion equation

$- \Delta u + u = f$

in the unit square $\Omega = (0,1)^2$ with source term

$f(x,y) = -5\exp\left(-(x-1/2)^2-(y-1/2)^2\right)$

and Dirichlet boundary conditions

$u = \begin{cases} \sin \pi x, &(x,y)\in\Gamma_{\rm inh}=\{(x,y)\in\partial\Omega\mid y=0\},\\ 0, &(x,y)\in\partial\Omega\setminus\Gamma_{\rm inh}. \end{cases}$

A suitable Dirichlet lift is

$g = \sin \pi x \; \cos \frac{\pi}{2} y.$

By setting $u = \tilde{u} + g$ the reaction-diffusion equation simplifies to a problem in $\tilde{u}$

$- \Delta \tilde{u} + \tilde{u} = f + \Delta g - g, \qquad x\in\Omega,$

with homogeneous Dirichlet boundary conditions

$\tilde{u} = 0, \qquad x\in\partial\Omega.$

Variational Formulation

Now we derive the corresponding variational formulation of the introduced problem: find $\tilde{u}\in H^1_0(\Omega)$ such that

$\int_{\Omega}\nabla\tilde{u}\cdot\nabla v\;{\rm d}x + \int_{\Omega}\tilde{u}v\;{\rm d}x = \int_{\Omega}fv\;{\rm d}x -\int_{\Omega}\nabla g\cdot\nabla v\;{\rm d}x - \int_{\Omega}gv\;{\rm d}x \qquad\forall v\in H^1_0(\Omega).$

Commented Program

First we have to import certain modules from the libconceptspy package.

from libconceptspy import concepts
from libconceptspy import hp1D
from libconceptspy import hp2D
from libconceptspy import graphics

The module concepts contains essential functionalities and must be loaded at first. The modules hp2D and includes all functions that are needed to build a representation of the hp-finite element space in two and dimensions respectively. The module graphics is used for the graphical output of the mesh and the solution.

In order to use MUMPS as a solver, we need to import MPI.

10from mpi4py import MPI

Main Program

The mesh is read from three files containing the coordinates, the elements and the boundary attributes of the mesh.

    try:
        mesh = concepts.Import2dMesh(coord   = SOURCEDIR + "/applications/unitsquareCoordinates.dat",
                                     elms    = SOURCEDIR + "/applications/unitsquareElements.dat",
                                     boundary= SOURCEDIR + "/applications/unitsquareEdges.dat"
                                  )
        print ('Mesh:')
        print (mesh)

If there is an error reading those files, then an exception error message is returned.

24 except RuntimeError:

25 print("Mesh import error")

else, the mesh is plotted using scaling factor of 100, a greyscale of 1.0 and one point per edge.

28 graphics.MeshEPS_r(msh = mesh, filename = "mesh.eps", scale=100, greyscale=1.0, nPoints=1)

The Dirichlet lift and its gradient are defined.

        DirichletLift = concepts.ParsedFormula_r(formula="(sin(pi*x)*cos(pi/2*y))")
        DirichletLiftGradx = concepts.ParsedFormula_r(formula="(pi*cos(pi*x)*cos(pi/2*y))")
        DirichletLiftGrady = concepts.ParsedFormula_r(formula="(-pi/2*sin(pi*x)*sin(pi/2*y))")

The problem for $\tilde{u}$ has homogeneous Dirichlet boundary conditions at all four edges.

        boundaryConditions = concepts.BoundaryConditions()
        for i in range(1,5): 
            boundaryConditions.add(attrib = i, bcObject = concepts.Boundary(type = concepts.Boundary.DIRICHLET))

Using the mesh and the homogeneous Dirichlet boundary conditions, the space can be built. We refine the space two times and set the polynomial degree to three. Then the elements of the space are built and the space is plotted.

        space = hp2D.hpAdaptiveSpaceH1 (msh = mesh, l = 2, p = 3, bc = boundaryConditions)
        space.rebuild();
        print('\nSpace:')
        print (space)
        graphics.MeshEPS_r(spc = space, filename = "space.eps", scale=100, greyscale=1.0, nPoints=1 )    

The right hand side is computed, containing the integrals of the source term, the Dirichlet lift and its gradient.

        lform = hp2D.Riesz_r(frm = concepts.ParsedFormula_r( formula = "(-5*exp(-(x-0.5)*(x-0.5)-(y-0.5)*(y-0.5)))" ) )
        rhs = concepts.Vector_r(spc = space, lf = lform)
 
        lformDirichletGrad = hp2D.GradLinearForm_r(frm1=DirichletLiftGradx, frm2=DirichletLiftGrady)
        rhsDirichletGrad =   concepts.Vector_r(spc = space, lf = lformDirichletGrad)
 
        lformDirichlet =  hp2D.Riesz_r(frm = DirichletLift)
        rhsDirichlet = concepts.Vector_r(spc = space, lf = lformDirichlet)
 
        rhs = rhs - rhsDirichletGrad - rhsDirichlet
        print('RHS Vector:\n')
        print(rhs)

The system matrix is computed from the bilinear form hp2D.Laplace_r and the bilinear form hp2D.Identity_r.

        la = hp2D.Laplace_r()
        A = concepts.SparseMatrix_r(spc = space, bf = la)
        A.compress()
        
        id = hp2D.Identity_r()
        M = concepts.SparseMatrix_r(spc = space, bf = id)
        M.compress()
        M.addInto(A,1.0)
        print('\n System Matrix:\n')
        print(A)

We solve the equation using a Mumps solver

        solver = concepts.Mumps_r(A = A)
        sol = concepts.Vector_r(spc = space)
        solver(fncY = rhs, fncX = sol)
        print('\nSolver:\n')
        print solver

In order to add the Dirichlet lift to the solution of the homogeneous Dirichlet problem we transform the solution, that is given as a vector related to the basis functions of space, into an element formula, that can be evaluated at each point in every cell.

        solFormula = concepts.ElementFormulaVector_1(spc = space, v = sol, f = hp2D.Value_r_r())
        print ('\nSolution:\n')
        print(concepts.ElementFormulaContainer_r_r(solFormula)+concepts.ElementFormulaContainer_r_r(DirichletLift))

We save the solutions, the homogeneous Dirichlet solution and the Dirichlet lift in a MAT-file. To this end, the shape functions are computed on equidistant points using the trapezoidal quadrature rule.

        hp2D.IntegrableQuad.setTensor( concepts.TRAPEZE, True, 8 )
        space.recomputeShapefunctions()
 
        # desired output
        graphics.MatlabBinaryGraphics(spc = space, 
                                      filename = "inhomDirichlet", 
                                      frm = concepts.ElementFormulaContainer_r_r(solFormula)+concepts.ElementFormulaContainer_r_r(DirichletLift))
        # output of homogeneous solution and Dirichlet lift
        graphics.MatlabBinaryGraphics(spc = space, filename = "inhomDirichlet0", frm = solFormula)
        graphics.MatlabBinaryGraphics(spc = space, filename = "inhomDirichletLift", frm = DirichletLift)

Results

Output of the program:

Mesh: Import2dMesh(ncell = 1)
 
RHS Vector:
Vector(121, [-7.813458e-01, -1.331759e-01, -1.446913e-04, -8.977796e-02, -1.975711e-02, -1.504745e-02, -1.103641e-04, -3.490666e-03,  4.167881e-05, -1.009906e+00, -1.735826e-01,  3.466376e-04, -1.628611e-01,  8.126026e-03, -2.796551e-02,  4.594644e-05, -1.436936e-03,  1.759809e-05, -8.586824e-01, -6.786590e-01, -1.598694e-01,  5.110721e-03, -1.388192e-01, -6.395889e-03, -1.248594e-01, -3.466673e-03, -2.580936e-02,  8.124231e-04, -1.244348e-03,  5.860425e-05, -8.122847e-02, -1.558898e-02, -1.461248e-02,  2.876690e-04, -3.028574e-03,  1.411786e-04, -7.813458e-01, -1.331759e-01, -1.446913e-04, -1.628611e-01,  8.126026e-03, -2.796551e-02,  4.594644e-05,  1.436936e-03, -1.759809e-05, -8.977796e-02, -1.975711e-02, -1.504745e-02, -1.103641e-04,  3.490666e-03, -4.167881e-05, -6.786590e-01, -8.122847e-02, -1.558898e-02, -1.248594e-01, -3.466673e-03, -1.461248e-02,  2.876690e-04,  3.028574e-03, -1.411786e-04, -1.388192e-01, -6.395889e-03, -2.580936e-02,  8.124231e-04,  1.244348e-03, -5.860425e-05, -4.822921e-01, -5.869802e-01, -9.922001e-02,  6.668103e-03, -9.533732e-02,  3.713134e-03, -1.236099e-01, -9.222017e-03, -2.002021e-02,  1.475922e-03,  8.660092e-04, -9.094635e-05, -6.180870e-02, -9.103448e-03, -1.221454e-02,  6.601945e-04,  2.115183e-03, -2.198685e-04, -6.016121e-02, -8.736539e-03, -8.218690e-03,  9.139240e-04,  8.895484e-04, -2.644021e-04, -7.032423e-02, -1.180407e-02, -1.147942e-02,  1.894019e-03,  3.595180e-04, -1.091848e-04, -4.822921e-01, -9.922001e-02,  6.668103e-03, -6.180870e-02, -9.103448e-03, -1.221454e-02,  6.601945e-04, -2.115183e-03,  2.198685e-04, -9.533732e-02,  3.713134e-03, -2.002021e-02,  1.475922e-03, -8.660092e-04,  9.094635e-05, -6.016121e-02, -8.736539e-03, -1.147942e-02,  1.894019e-03, -3.595180e-04,  1.091848e-04, -8.218690e-03,  9.139240e-04, -8.895484e-04,  2.644021e-04])
 
System Matrix:
SparseMatrix(121x121, HashedSparseMatrix: 1521 (10.3886%) entries bound.)
 
Solver:
Mumps(n = 121)
 
Solution:
FrmE_Sum(ElementFormulaVector<1>(hp2D::Value<double>(), Vector(121, [-5.387654e-01, -3.006218e-01,  1.564124e-02, -1.425659e-01, -4.809499e-03, -1.088256e-01,  1.611971e-02,  8.625588e-03, -6.891406e-03, -7.373583e-01, -4.048944e-01,  1.925875e-02, -1.941122e-01,  2.767690e-03, -1.016416e-01,  3.259733e-03, -1.569518e-03,  2.857100e-04, -8.461932e-01, -6.183710e-01, -2.393483e-01,  9.380839e-03, -2.230022e-01, -2.981412e-03, -1.722831e-01, -6.882467e-03, -6.528885e-02,  2.490903e-03, -1.108972e-03, -1.050398e-05, -1.618117e-01, -6.560393e-03, -3.530271e-02,  1.727733e-03, -4.305751e-03,  3.179923e-04, -5.387655e-01, -3.006218e-01,  1.564124e-02, -1.941122e-01,  2.767687e-03, -1.016416e-01,  3.259733e-03,  1.569518e-03, -2.857100e-04, -1.425659e-01, -4.809502e-03, -1.088256e-01,  1.611971e-02, -8.625588e-03,  6.891407e-03, -6.183710e-01, -1.618117e-01, -6.560394e-03, -1.722831e-01, -6.882469e-03, -3.530271e-02,  1.727733e-03,  4.305752e-03, -3.179924e-04, -2.230022e-01, -2.981412e-03, -6.528885e-02,  2.490903e-03,  1.108972e-03,  1.050400e-05, -4.161721e-01, -5.639852e-01, -1.159405e-01,  2.828855e-03, -1.454700e-01,  1.444007e-03, -1.596678e-01, -4.353042e-03, -4.288914e-02,  1.375478e-03,  4.954845e-04, -1.172436e-04, -1.224176e-01, -1.613845e-03, -2.602445e-02, -1.262254e-04,  2.824662e-03,  6.029993e-05, -1.041387e-01,  1.097789e-03, -7.650684e-02, -1.372658e-02, -1.375732e-02, -7.269106e-03, -1.270253e-01, -1.308806e-03, -2.361722e-02,  2.517779e-03, -5.561139e-04,  1.292617e-04, -4.161721e-01, -1.159405e-01,  2.828853e-03, -1.224176e-01, -1.613847e-03, -2.602445e-02, -1.262255e-04, -2.824662e-03, -6.029997e-05, -1.454700e-01,  1.444005e-03, -4.288914e-02,  1.375478e-03, -4.954845e-04,  1.172437e-04, -1.041387e-01,  1.097788e-03, -2.361722e-02,  2.517779e-03,  5.561139e-04, -1.292618e-04, -7.650684e-02, -1.372658e-02,  1.375732e-02,  7.269106e-03])) + ParsedFormula<Real>((sin(pi*x)*cos(pi/2*y))))

Matlab plot of the solution: Matlab output

@section complete Complete Source Code