For more information and detailed explain of this example see Solving the Poisson equation

* \file analytical_poisson.cpp
* \ingroup mofem_simple_interface
* \example analytical_poisson.cpp
* For more information and detailed explain of this
* example see \ref poisson_tut1
/* This file is part of MoFEM.
* MoFEM is free software: you can redistribute it and/or modify it under
* the terms of the GNU Lesser General Public License as published by the
* Free Software Foundation, either version 3 of the License, or (at your
* option) any later version.
* MoFEM is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* License for more details.
* You should have received a copy of the GNU Lesser General Public
* License along with MoFEM. If not, see <http://www.gnu.org/licenses/>. */
#include <PoissonOperators.hpp>
static char help[] = "...\n\n";
* \brief Function
* This is prescribed exact function. If this function is given by polynomial
* order of "p" and order of approximation is "p" or higher, solution of
* finite element method is exact (with machine precision).
* \f[
* u = 1+x^2+y^2+z^3
* \f]
struct ExactFunction {
double operator()(const double x, const double y, const double z) const {
return 1 + x * x + y * y + z * z * z;
* \brief Exact gradient
FTensor::Tensor1<double, 3> operator()(const double x, const double y,
const double z) const {
grad(0) = 2 * x;
grad(1) = 2 * y;
grad(2) = 3 * z * z;
return grad;
* \brief Laplacian of function.
* This is Laplacian of \f$u\f$, it is calculated using formula
* \f[
* \nabla^2 u(x,y,z) = \nabla \cdot \nabla u
* \frac{\partial^2 u}{\partial x^2}+
* \frac{\partial^2 u}{\partial y^2}+
* \frac{\partial^2 u}{\partial z^2}
* \f]
double operator()(const double x, const double y, const double z) const {
return 4 + 6 * z;
int main(int argc, char *argv[]) {
// Initialize PETSc
MoFEM::Core::Initialize(&argc,&argv,(char *)0,help);
try {
// Create MoAB database
moab::Core moab_core; // create database
moab::Interface &moab = moab_core; // create interface to database
// Get command line options
int order = 3; // default approximation order
PetscBool flg_test = PETSC_FALSE; // true check if error is numerical error
CHKERR PetscOptionsBegin(PETSC_COMM_WORLD, "", "Poisson's problem options",
// Set approximation order
CHKERR PetscOptionsInt("-order", "approximation order", "", order, &order,
// Set testing (used by CTest)
CHKERR PetscOptionsBool("-test", "if true is ctest", "", flg_test,
&flg_test, PETSC_NULL);
ierr = PetscOptionsEnd();
// Create MoFEM database and link it to MoAB
MoFEM::Core mofem_core(moab); // create database
MoFEM::Interface &m_field = mofem_core; // create interface to database
// Register DM Manager
CHKERR DMRegister_MoFEM("DMMOFEM"); // register MoFEM DM in PETSc
// Create vector to store approximation global error
Vec global_error;
// First we crate elements, implementation of elements is problem independent,
// we do not know yet what fields are present in the problem, or
// even we do not decided yet what approximation base or spaces we
// are going to use. Implementation of element is free from
// those constrains and can be used in different context.
// Elements used by KSP & DM to assemble system of equations
boost::shared_ptr<ForcesAndSourcesCore> domain_lhs_fe; ///< Volume element for the matrix
boost::shared_ptr<ForcesAndSourcesCore> boundary_lhs_fe; ///< Boundary element for the matrix
boost::shared_ptr<ForcesAndSourcesCore> domain_rhs_fe; ///< Volume element to assemble vector
boost::shared_ptr<ForcesAndSourcesCore> boundary_rhs_fe; ///< Volume element to assemble vector
boost::shared_ptr<ForcesAndSourcesCore> domain_error; ///< Volume element evaluate error
boost::shared_ptr<ForcesAndSourcesCore> post_proc_volume; ///< Volume element to Post-process results
boost::shared_ptr<ForcesAndSourcesCore> null; ///< Null element do nothing
// Add problem specific operators the generic finite elements to calculate matrices and vectors.
boundary_lhs_fe, domain_rhs_fe, boundary_rhs_fe);
// Add problem specific operators the generic finite elements to calculate error on elements and global error
// in H1 norm
global_error, domain_error);
// Post-process results
// Get simple interface is simplified version enabling quick and
// easy construction of problem.
Simple *simple_interface;
// Query interface and get pointer to Simple interface
CHKERR m_field.getInterface(simple_interface);
// Build problem with simple interface
// Get options for simple interface from command line
CHKERR simple_interface->getOptions();
// Load mesh file to database
CHKERR simple_interface->loadFile();
// Add field on domain and boundary. Field is declared by space and base and rank. space
// can be H1. Hcurl, Hdiv and L2, base can be AINSWORTH_LEGENDRE_BASE, DEMKOWICZ_JACOBI_BASE and more,
// where rank is number of coefficients for dof.
// Simple interface assumes that there are four types of field; 1) defined
// on body domain, 2) fields defined on body boundary, 3) skeleton field defined
// on finite element skeleton. Finally data field is defined on body domain. Data field
// is not solved but used for post-process or to keep material parameters, etc. Here
// we using it to calculate approximation error on elements.
// Add domain filed "U" in space H1 and Legendre base, Ainsworth recipe is used
// to construct base functions.
CHKERR simple_interface->addDomainField("U",H1,AINSWORTH_LEGENDRE_BASE,1);
// Add Lagrange multiplier field on body boundary
CHKERR simple_interface->addBoundaryField("L",H1,AINSWORTH_LEGENDRE_BASE,1);
// Add error (data) field, we need only L2 norm. Later order is set to 0, so this
// is piecewise discontinuous constant approx., i.e. 1 DOF for element. You can use
// more DOFs and collate moments of error to drive anisotropic h/p-adaptivity, however
// this is beyond this example.
CHKERR simple_interface->addDataField("ERROR",L2,AINSWORTH_LEGENDRE_BASE,1);
// Set fields order domain and boundary fields.
CHKERR simple_interface->setFieldOrder("U",order); // to approximate function
CHKERR simple_interface->setFieldOrder("L",order); // to Lagrange multipliers
CHKERR simple_interface->setFieldOrder("ERROR",0); // approximation order for error
// Setup problem. At that point database is constructed, i.e. fields, finite elements and
// problem data structures with local and global indexing.
CHKERR simple_interface->setUp();
// Get access to PETSC-MoFEM DM manager.
// or more derails see <http://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/DM/index.html>
// Form that point internal MoFEM data structures are linked with PETSc interface. If
// DM functions contains string MoFEM is is MoFEM specific DM interface function,
// otherwise DM function part of standard PETSc interface.
DM dm;
// Get dm
CHKERR simple_interface->getDM(&dm);
// Set KSP context for DM. At that point only elements are added to DM operators.
// Calculations of matrices and vectors is executed by KSP solver. This part
// of the code makes connection between implementation of finite elements and
// data operators with finite element declarations in DM manager. From that
// point DM takes responsibility for executing elements, calculations of
// matrices and vectors, and solution of the problem.
// Set operators for KSP solver
dm, simple_interface->getDomainFEName(), domain_lhs_fe, null, null);
dm, simple_interface->getBoundaryFEName(), boundary_lhs_fe, null,
// Set calculation of the right hand side vector for KSP solver
CHKERR DMMoFEMKSPSetComputeRHS(dm, simple_interface->getDomainFEName(),
domain_rhs_fe, null, null);
CHKERR DMMoFEMKSPSetComputeRHS(dm, simple_interface->getBoundaryFEName(),
boundary_rhs_fe, null, null);
// Solve problem, only PETEc interface here.
// Create the right hand side vector and vector of unknowns
Vec F,D;
CHKERR DMCreateGlobalVector(dm,&F);
// Create unknown vector by creating duplicate copy of F vector. only
// structure is duplicated no values.
CHKERR VecDuplicate(F,&D);
// Create solver and link it to DM
KSP solver;
CHKERR KSPSetFromOptions(solver);
CHKERR KSPSetDM(solver,dm);
// Set-up solver, is type of solver and pre-conditioners
CHKERR KSPSetUp(solver);
// At solution process, KSP solver using DM creates matrices, Calculate
// values of the left hand side and the right hand side vector. then
// solves system of equations. Results are stored in vector D.
CHKERR KSPSolve(solver,F,D);
// Scatter solution on the mesh. Stores unknown vector on field on the mesh.
// Clean data. Solver and vector are not needed any more.
CHKERR KSPDestroy(&solver);
CHKERR VecDestroy(&D);
CHKERR VecDestroy(&F);
// Calculate error
// Loop over all elements in mesh, and run users operators on each element.
CHKERR DMoFEMLoopFiniteElements(dm, simple_interface->getDomainFEName(),
if (flg_test == PETSC_TRUE) {
// Loop over all elements in the mesh and for each execute post_proc_volume
// element and operators on it.
CHKERR DMoFEMLoopFiniteElements(dm, simple_interface->getDomainFEName(),
// Write results
CHKERR boost::static_pointer_cast<PostProcVolumeOnRefinedMesh>(
// Destroy DM, no longer needed.
CHKERR DMDestroy(&dm);
// Destroy ghost vector
CHKERR VecDestroy(&global_error);
// finish work cleaning memory, getting statistics, etc.
return 0;