Operator.h

// ============================================================================
// ==                                                                        ==
// == AMDiS - Adaptive multidimensional simulations                          ==
// ==                                                                        ==
// ============================================================================
// ==                                                                        ==
// ==  crystal growth group                                                  ==
// ==                                                                        ==
// ==  Stiftung caesar                                                       ==
// ==  Ludwig-Erhard-Allee 2                                                 ==
// ==  53175 Bonn                                                            ==
// ==  germany                                                               ==
// ==                                                                        ==
// ============================================================================
// ==                                                                        ==
// ==  http://www.caesar.de/cg/AMDiS                                         ==
// ==                                                                        ==
// ============================================================================

/** \file Operator.h */

#ifndef AMDIS_OPERATOR_H
#define AMDIS_OPERATOR_H

#include <vector>
#include "AMDiS_fwd.h"
#include "FixVec.h"
#include "Flag.h"
#include "MatrixVector.h"
#include "ElInfo.h"
#include "AbstractFunction.h"
#include "OpenMP.h"
#include "SubAssembler.h"

namespace AMDiS {

  /** 
   * \ingroup Assembler
   * 
   * \brief
   * Base class for ZeroOrderTerm, FirstOrderTerm and SecondOrderTerm. 
   * OperatorTerms are the building blocks of an Operator. Each OperatorTerm
   * has its properties which are regarded, when constructing 
   * an Assembler for the corresponding Operator.
   */
  class OperatorTerm
  {
  public:
    /** \brief
     * Constructs an OperatorTerm with initially no properties.
     * degree_ is used to determine the degree of the needed quadrature
     * for the assemblage.  
     */
    OperatorTerm(int deg) 
      : properties(0), 
	degree(deg),
	auxFESpaces(0)
    {}

    /// Destructor.
    virtual ~OperatorTerm() {}

    /** \brief
     * Virtual method. It's called by SubAssembler::initElement() for
     * each OperatorTerm belonging to this SubAssembler. E.g., vectors
     * and coordinates at quadrature points can be calculated here.
     */
    virtual void initElement(const ElInfo*, 
			     SubAssembler*,
			     Quadrature *quad = NULL) 
    {}

    virtual void initElement(const ElInfo* largeElInfo,
			     const ElInfo* smallElInfo,
			     SubAssembler*,
			     Quadrature *quad = NULL)
    {}

    /// Returs \auxFESpaces, the list of all aux fe spaces the operator makes use off.
    std::vector<const FiniteElemSpace*>& getAuxFESpaces() 
    {
      return auxFESpaces;
    }

    /// Specifies whether the matrix of the term is symmetric
    void setSymmetric(bool symm);

    /// Returns true, if the term is piecewise constant, returns false otherwise
    inline bool isPWConst() 
    { 
      return (degree == 0);
    }

    /// Returns true, if the term has a symmetric matrix, returns false otherwise.
    bool isSymmetric();

    /// Returns \ref degree.
    inline int getDegree() 
    { 
      return degree; 
    }

    /// Evaluation of the OperatorTerm at all quadrature points.
    virtual void eval(int nPoints,
		      const double *uhAtQP,
		      const WorldVector<double> *grdUhAtQP,
		      const WorldMatrix<double> *D2UhAtQP,
		      double *result,
		      double factor) const = 0;

    /** \brief
     * Determines the value of a dof vector at the quadrature points of a given 
     * element. It is used by all VecAtQP like operator terms.
     */
    double *getVectorAtQPs(DOFVectorBase<double>* vec,
			   const ElInfo* elInfo, 
			   SubAssembler* subAssembler,
			   Quadrature *quad);		   

    /** \brief
     * Determines the value of a dof vector at the quadrature points of a given 
     * element. This function is used, if an operator is assembled on two different
     * meshes using the dual traverse. The element, i.e. the small or the large one,
     * is choosen, which corresponds to the mesh the dof vector is defined on.
     */
    double *getVectorAtQPs(DOFVectorBase<double>* vec,
			   const ElInfo* smallElInfo, 
			   const ElInfo* largeElInfo, 
			   SubAssembler* subAssembler,
			   Quadrature *quad);	

    ///
    WorldVector<double>* getGradientsAtQPs(DOFVectorBase<double>* vec,
					   const ElInfo* elInfo,
					   SubAssembler* subAssembler,
					   Quadrature *quad);

    ///
    WorldVector<double>* getGradientsAtQPs(DOFVectorBase<double>* vec,
					   const ElInfo* smallElInfo, 
					   const ElInfo* largeElInfo,
					   SubAssembler* subAssembler,
					   Quadrature *quad);

  protected:
    /// Evaluation of \f$ \Lambda \cdot A \cdot \Lambda^t\f$.
    static void lalt(const DimVec<WorldVector<double> >& Lambda,
		     const WorldMatrix<double>& matrix,
		     DimMat<double>& LALt,
		     bool symm,
		     double factor);

    /** \brief
     * Evaluation of \f$ \Lambda \cdot A \cdot \Lambda^t\f$ for \f$ A \f$
     * the matrix having a ONE in the position \f$ (K,L) \f$
     * and ZEROS in all other positions.
     */
    static void lalt_kl(const DimVec<WorldVector<double> >& Lambda,
			int k, int l,
			DimMat<double>& LALt,
			double factor);

    /// Evaluation of \f$ \Lambda \cdot A \cdot \Lambda^t\f$ for A equal to the identity.
    static void l1lt(const DimVec<WorldVector<double> >& Lambda, 
		     DimMat<double>& LALt,
		     double factor);

    /// Evaluation of \f$ \Lambda \cdot b\f$.
    static inline void lb(const DimVec<WorldVector<double> >& Lambda,
			  const WorldVector<double>& b,
			  DimVec<double>& Lb,
			  double factor)
    {
      const int dimOfWorld = Global::getGeo(WORLD);
      
      for (int i = 0; i <= dimOfWorld; i++) {
	double val = 0.0;
	
	for (int j = 0; j < dimOfWorld; j++)
	  val += Lambda[i][j] * b[j];
	
	Lb[i] += val * factor;
      }    
    }

    /** \brief
     * Evaluation of \f$ \Lambda \cdot b\f$ if b contains the value 1.0 in 
     * each component.
     */
    static inline void l1(const DimVec<WorldVector<double> >& Lambda,
			  DimVec<double>& Lb,
			  double factor)
    {
      const int dimOfWorld = Global::getGeo(WORLD);

      for (int i = 0; i <= dimOfWorld; i++) {
	double val = 0.0;
      
	for (int j = 0; j < dimOfWorld; j++)
	  val += Lambda[i][j];

	Lb[i] += val * factor;
      }    
    }

  protected:
    /// Stores the properties of this OperatorTerm
    Flag properties;

    /// Polynomial degree of the term. Used to detemine the degree of the quadrature.
    int degree;

    /// List off all fe spaces, the operator term makes use off.
    std::vector<const FiniteElemSpace*> auxFESpaces;

    /// Pointer to the Operator this OperatorTerm belongs to.
    Operator* operat;

    /// Flag for piecewise constant terms
    static const Flag PW_CONST;

    /// Flag for symmetric terms
    static const Flag SYMMETRIC;

    friend class SubAssembler;
    friend class ZeroOrderAssembler;
    friend class FirstOrderAssembler;
    friend class SecondOrderAssembler;
    friend class Operator;
  };

  /**
   * \ingroup Assembler
   * 
   * \brief
   * Describes the second order terms: \f$ \nabla \cdot A \nabla u(\vec{x}) \f$
   */
  class SecondOrderTerm : public OperatorTerm
  {
  public:
    /// Constructor.
    SecondOrderTerm(int deg) 
      : OperatorTerm(deg) 
    {}

    /// Destructor.
    virtual ~SecondOrderTerm() 
    {}

    /// Evaluation of \f$ \Lambda A \Lambda^t \f$ at all quadrature points.
    virtual void getLALt(const ElInfo *elInfo, 
			 int nPoints, 
			 DimMat<double> **result) const = 0;

    /// Evaluation of \f$ A \nabla u(\vec{x}) \f$ at all quadrature points.
    virtual void weakEval(int nPoints,
			  const WorldVector<double> *grdUhAtQP,
			  WorldVector<double> *result) const = 0;

  };

  /**
   * \ingroup Assembler
   * 
   * \brief
   * Implements the laplace operator: \f$ \Delta u(\vec{x}) \f$
   */
  class Laplace_SOT : public SecondOrderTerm 
  {
  public:
    /// Constructor.
    Laplace_SOT() 
      : SecondOrderTerm(0) 
    {
      setSymmetric(true);
    }

    /// Implenetation of SecondOrderTerm::getLALt().
    inline void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const
    {
      const DimVec<WorldVector<double> > &Lambda = elInfo->getGrdLambda();

      for (int iq = 0; iq < nPoints; iq++)
	l1lt(Lambda, *(LALt[iq]), 1.0);
    }

    /// Implementation of SecondOrderTerm::eval().
    inline void eval(int nPoints,
		     const double *,
		     const WorldVector<double> *,
		     const WorldMatrix<double> *D2UhAtQP,
		     double *result,
		     double factor) const
    {
      int dow = Global::getGeo(WORLD);
    
      if (D2UhAtQP) {
	for (int iq = 0; iq < nPoints; iq++) {
	  double resultQP = 0.0;
	  for (int i = 0; i < dow; i++) 
	    resultQP += D2UhAtQP[iq][i][i];
	  result[iq] += factor * resultQP;
	}
      }
    }

    /// Implenetation of SecondOrderTerm::weakEval().
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const
    {
      if (grdUhAtQP)
	for (int iq = 0; iq < nPoints; iq++)
	  result[iq] += grdUhAtQP[iq];
    }
  };

  /**
   * \ingroup Assembler
   * 
   * \brief
   * Implements the laplace operator multiplied with a scalar factor:
   * \f$ f \cdot \Delta u(\vec{x}) \f$
   */
  class FactorLaplace_SOT : public SecondOrderTerm 
  {
  public:
    /// Constructor.
    FactorLaplace_SOT(double f) 
      : SecondOrderTerm(0)  
    {
      factor = new double;
      *factor = f;

      setSymmetric(true);
    }

    /// Constructor.
    FactorLaplace_SOT(double *fptr) 
      : SecondOrderTerm(0), 
	factor(fptr)
    {
      setSymmetric(true);
    }

    /// Implements SecondOrderTerm::getLALt().
    inline void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const
    {
      const DimVec<WorldVector<double> > &Lambda = elInfo->getGrdLambda();
      for (int iq = 0; iq < nPoints; iq++) 
	l1lt(Lambda, *(LALt[iq]), (*factor));
    }

    /// Implenetation of SecondOrderTerm::eval().
    void eval(int nPoints,
	      const double *,
	      const WorldVector<double> *,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double f) const
    {
      int dow = Global::getGeo(WORLD);

      if (D2UhAtQP) {
	for (int iq = 0; iq < nPoints; iq++) {
	  double resultQP = 0.0;
	  for (int i = 0; i < dow; i++) {
	    resultQP += D2UhAtQP[iq][i][i];
	  }
	  result[iq] += resultQP * f * (*factor);
	}
      }
    }

    /// Implenetation of SecondOrderTerm::weakEval().
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const
    {
      if (grdUhAtQP)
	for (int iq = 0; iq < nPoints; iq++)
	  axpy(*factor, grdUhAtQP[iq], result[iq]);
    }

  private:
    /// Pointer to the factor.
    double *factor;
  };

  /**
   * \ingroup Assembler
   * 
   * \brief
   * SecondOrderTerm where A is a function which maps a DOFVector evaluated at
   * a given quadrature point to a WolrdMatrix:
   * \f$ \nabla \cdot A(v(\vec{x})) \nabla u(\vec{x}) \f$
   */
  class MatrixFct_SOT : public SecondOrderTerm 
  {
  public:
    /// Constructor.
    MatrixFct_SOT(DOFVectorBase<double> *dv, 
		  AbstractFunction<WorldMatrix<double>, double> *fct,
		  AbstractFunction<WorldVector<double>, WorldMatrix<double> > *div,
		  bool sym = false);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;
  
    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    /** \brief
     * DOFVector to be evaluated at quadrature points.
     */
    DOFVectorBase<double>* vec;

    /** \brief
     * Pointer to the values of the DOFVector at quadrature points.
     */
    double* vecAtQPs;

    /** \brief
     * Function for A.
     */
    AbstractFunction<WorldMatrix<double>, double>* matrixFct;

    /** \brief
     *
     */
    AbstractFunction<WorldVector<double>, WorldMatrix<double> >* divFct;

    /** \brief
     * True, if \ref matrixFct produces always symmetric matrices.
     */
    bool symmetric;
  };

  /** 
   * \ingroup Assembler
   *
   * \brief
   * SecondOrderTerm where A is a given fixed WorldMatrix<double>:
   * \f$ \nabla \cdot A \nabla u(\vec{x}) \f$
   */
  class Matrix_SOT : public SecondOrderTerm {
  public:
    /// Constructor
    Matrix_SOT(WorldMatrix<double> mat) 
      : SecondOrderTerm(0), matrix(mat)
    {
      symmetric = matrix.isSymmetric();
      setSymmetric(symmetric);
    }

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    inline void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const
    {
      const DimVec<WorldVector<double> >& Lambda = elInfo->getGrdLambda();
      for (int iq = 0; iq < nPoints; iq++) 
	lalt(Lambda, matrix, *(LALt[iq]), symmetric, 1.0);
    }
  
    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    /** \brief
     * Matrix stroring A.
     */
    WorldMatrix<double> matrix;

    /** \brief
     * True, if \ref matrix is symmetric.
     */
    bool symmetric;
  };

  /** 
   * \ingroup Assembler
   *
   * \brief
   * SecondOrderTerm where A is a WorldMatrix<double> having a ONE in position IJ
   * and ZERO in all other positions
   * \f$ \nabla \cdot A \nabla u(\vec{x}) \f$
   */
  class FactorIJ_SOT : public SecondOrderTerm 
  {
  public:
    /// Constructor.
    FactorIJ_SOT(int x_i, int x_j, double f) 
      : SecondOrderTerm(0), xi(x_i), xj(x_j)
    {
      factor = new double;
      *factor = f;

      setSymmetric(xi == xj);
    }

    /// Constructor.
    FactorIJ_SOT(int x_i, int x_j, double *fptr) 
      : SecondOrderTerm(0), xi(x_i), xj(x_j), factor(fptr)
    {
      setSymmetric(xi == xj);
    }

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    inline void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const
    {
      const DimVec<WorldVector<double> > &Lambda     = elInfo->getGrdLambda();
      for (int iq = 0; iq < nPoints; iq++)
	lalt_kl(Lambda, xi, xj, *(LALt[iq]), (*factor));
    }

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double              *,
	      const WorldVector<double> *,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double fac) const
    {
      if (D2UhAtQP)
	for (int iq = 0; iq < nPoints; iq++)
	  result[iq] += (*factor) * D2UhAtQP[iq][xi][xj] * fac;
    }

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const
    {
      if (grdUhAtQP)
	for (int iq = 0; iq < nPoints; iq++)
	  result[iq][xi] += (*factor) * grdUhAtQP[iq][xj];
    }

  private:
    /** \brief
     * Directions for the derivatives.
     */
    int xi, xj;

    /** \brief
     * Pointer to the factor.
     */
    double *factor;
  };

  /**
   * \ingroup Assembler
   *  
   * \brief
   * Laplace operator multiplied with a function evaluated at the quadrature
   * points of a given DOFVector:
   * \f$ f(v(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class VecAtQP_SOT : public SecondOrderTerm {
  public:
    /// Constructor.
    VecAtQP_SOT(DOFVectorBase<double> *dv, AbstractFunction<double, double> *af);

    /// Implementation of \ref OperatorTerm::initElement().
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /// Implementation of \ref OperatorTerm::initElement() for multilpe meshes.
    void initElement(const ElInfo* smallElInfo,
		     const ElInfo* largeElInfo,
		     SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /// Implements SecondOrderTerm::getLALt().
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;    

    /// Implenetation of SecondOrderTerm::eval().
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /// Implenetation of SecondOrderTerm::weakEval().
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    /// DOFVector to be evaluated at quadrature points.
    DOFVectorBase<double>* vec;

    /// Pointer to an array containing the DOFVector evaluated at quadrature points.
    double* vecAtQPs;

    /// Function evaluated at \ref vecAtQPs.
    AbstractFunction<double, double> *f;
  };

  /**
   * \ingroup Assembler
   *  
   * \brief
   * Laplace operator multiplied with a function evaluated at the quadrature
   * points of a given DOFVector:
   * \f$ f(v(\vec{x}), w(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class Vec2AtQP_SOT : public SecondOrderTerm {
  public:
    /// Constructor.
    Vec2AtQP_SOT(DOFVectorBase<double> *dv1, DOFVectorBase<double> *dv2, 
		 BinaryAbstractFunction<double, double, double> *af);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;    
    
    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;
    
    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;
    
  protected:
    /** \brief
     * DOFVector to be evaluated at quadrature points.
     */
    DOFVectorBase<double>* vec1;
    DOFVectorBase<double>* vec2;
    
    /** \brief
     * Pointer to an array containing the DOFVector evaluated at quadrature
     * points.
     */
    double* vecAtQPs1;
    double* vecAtQPs2;
    
    /** \brief
     * Function evaluated at \ref vecAtQPs.
     */
    BinaryAbstractFunction<double, double, double> *f;
  };

  /**
   * \ingroup Assembler
   * 
   * \brief
   * Laplace multiplied with a function evaluated at each quadrature point:
   * \f$ f(\vec{x}) \Delta u(\vec{x}) \f$
   */
  class CoordsAtQP_SOT : public SecondOrderTerm {
  public:
    /// Constructor.
    CoordsAtQP_SOT(AbstractFunction<double, WorldVector<double> > *af)
      : SecondOrderTerm(af->getDegree()), g(af)
    {
      setSymmetric(true);
    }

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;    

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;


  protected:
    /** \brief
     * Stores coordinates at quadrature points. Set in \ref initElement().
     */
    WorldVector<double>* coordsAtQPs;

    /** \brief
     * Function evaluated at quadrature points.
     */
    AbstractFunction<double, WorldVector<double> > *g;
  };

  /**
   * \ingroup Assembler
   *
   * \brief
   * SecondOrderTerm where A is a function which maps the gradient of a 
   * DOFVector at each quadrature point to WorldMatrix<double>:
   * \f$ \nabla \cdot A(\nabla v(\vec{x})) \nabla u(\vec{x})\f$
   */
  class MatrixGradient_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    MatrixGradient_SOT(DOFVectorBase<double> *dv,
		       AbstractFunction<WorldMatrix<double>, WorldVector<double> > *af,
		       AbstractFunction<WorldVector<double>, WorldMatrix<double> > *divAf,
		       bool symm = false);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    DOFVectorBase<double>* vec;

    /** \brief
     * Function which maps each entry in \ref gradAtQPs to a WorldMatrix<double>.
     */
    AbstractFunction<WorldMatrix<double>, WorldVector<double> > *f;

    AbstractFunction<WorldVector<double>, WorldMatrix<double> > *divFct;

    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    WorldVector<double>* gradAtQPs;

    /** \brief
     * True, if \ref f provides always a symmetric WorldMatrix<double>.
     */
    bool symmetric;
  };

  /**
   * \ingroup Assembler
   *
   * \brief
   * Laplace multiplied with a function which maps the gradient of a DOFVector
   * at each quadrature point to a double:
   * \f$ f(\nabla v(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class FctGradient_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    FctGradient_SOT(DOFVectorBase<double> *dv,  
		    AbstractFunction<double, WorldVector<double> > *af);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    DOFVectorBase<double>* vec;

    /** \brief
     * Function wich maps \ref gradAtQPs to a double.
     */
    AbstractFunction<double, WorldVector<double> > *f;

    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    WorldVector<double>* gradAtQPs;
  };


  /**
   * \ingroup Assembler
   *
   * \brief
   * Laplace multiplied with a function which maps the gradient of a DOFVector
   * at each quadrature point to a double:
   * \f$ f(\nabla v(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class VecAndGradientAtQP_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    VecAndGradientAtQP_SOT(DOFVectorBase<double> *dv,
			   BinaryAbstractFunction<double, double, WorldVector<double> > *af);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    DOFVectorBase<double>* vec;

    /** \brief
     * Function wich maps \ref gradAtQPs to a double.
     */
    BinaryAbstractFunction<double, double, WorldVector<double> > *f;

    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    double* vecAtQPs;
    WorldVector<double>* gradAtQPs;
  };

  /**
   * \ingroup Assembler
   *
   * \brief
   * Laplace multiplied with a function which maps the gradient of a DOFVector
   * at each quadrature point to a double:
   * \f$ \nabla \cdot A(v(\vec{x}), \nabla v(\vec{x})) \nabla u(\vec{x}) \f$
   */
  class VecMatrixGradientAtQP_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    VecMatrixGradientAtQP_SOT(DOFVectorBase<double> *dv,
			      BinaryAbstractFunction<WorldMatrix<double>, double, WorldVector<double> > *af,
			      AbstractFunction<WorldVector<double>, WorldMatrix<double> > *divAf,
			      bool symm = false);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;
    
  protected:
    DOFVectorBase<double>* vec;
    
    /** \brief
     * Function wich maps \ref gradAtQPs to a double.
     */
    BinaryAbstractFunction<WorldMatrix<double>, 
			   double, WorldVector<double> > *f;
    
    AbstractFunction<WorldVector<double>, WorldMatrix<double> > *divFct;
    
    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    double* vecAtQPs;
    WorldVector<double>* gradAtQPs;
    
    bool symmetric;
  };

  /**
   * \ingroup Assembler
   *
   * \brief
   * Laplace multiplied with a function which maps the gradient of a DOFVector
   * at each quadrature point to a double:
   * \f$ f(\nabla v(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class VecGradCoordsAtQP_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    VecGradCoordsAtQP_SOT(DOFVectorBase<double> *dv,
			  TertiaryAbstractFunction<double, double,
			  WorldVector<double>, WorldVector<double> > *af);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    DOFVectorBase<double>* vec;

    /** \brief
     * Function wich maps \ref gradAtQPs to a double.
     */
    TertiaryAbstractFunction<double, double, WorldVector<double>,
			     WorldVector<double> > *f;

    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    double* vecAtQPs;
    WorldVector<double>* gradAtQPs;
    WorldVector<double>* coordsAtQPs;
  
  };


  /**
   * \ingroup Assembler
   *  
   * \brief
   * Laplace operator multiplied with a function evaluated at the quadrature
   * points of a given DOFVector:
   * \f$ -f(v(\vec{x})) \Delta u(\vec{x}) \f$
   */
  class VecAndCoordsAtQP_SOT : public SecondOrderTerm {
  public:
    /// Constructor.
    VecAndCoordsAtQP_SOT(DOFVectorBase<double> *dv, 
			 BinaryAbstractFunction<double, double, WorldVector<double> > *af);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::eval().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;    

    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    /** \brief
     * DOFVector to be evaluated at quadrature points.
     */
    DOFVectorBase<double>* vec;

    /** \brief
     * Pointer to an array containing the DOFVector evaluated at quadrature
     * points.
     */
    double* vecAtQPs;
  
    WorldVector<double>* coordsAtQPs;

    /** \brief
     * Function evaluated at \ref vecAtQPs.
     */
    BinaryAbstractFunction<double, double,  WorldVector<double> > *f;
  };

  /**
   * \ingroup Assembler
   *
   * \brief
   * SecondOrderTerm where A is a function which maps the gradient of a 
   * DOFVector at each quadrature point to WorldMatrix<double>:
   * \f$ \nabla \cdot A(\nabla v(\vec{x})) \nabla u(\vec{x})\f$
   */
  class MatrixGradientAndCoords_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    MatrixGradientAndCoords_SOT(DOFVectorBase<double> *dv,
				BinaryAbstractFunction<WorldMatrix<double>,
				WorldVector<double> , 
				WorldVector<double> > *af,
				AbstractFunction<WorldVector<double>,
				WorldMatrix<double> > *divAf,
				bool symm = false);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo* elInfo, SubAssembler* subAssembler,
		     Quadrature *quad = NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    DOFVectorBase<double>* vec;

    /** \brief
     * Function which maps each entry in \ref gradAtQPs to a WorldMatrix<double>.
     */
    BinaryAbstractFunction<WorldMatrix<double>,
			   WorldVector<double>, WorldVector<double> > *f;

    AbstractFunction<WorldVector<double>, WorldMatrix<double> > *divFct;

    /** \brief
     * Pointer to a WorldVector<double> array containing the gradients of the DOFVector
     * at each quadrature point.
     */
    WorldVector<double>* gradAtQPs;

    WorldVector<double>* coordsAtQPs;

    /** \brief
     * True, if \ref f provides always a symmetric WorldMatrix<double>.
     */
    bool symmetric;
  };

  class General_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    General_SOT(std::vector<DOFVectorBase<double>*> vecs,
		std::vector<DOFVectorBase<double>*> grads,
		TertiaryAbstractFunction<WorldMatrix<double>, 
		  WorldVector<double>,
		  std::vector<double>, 
		  std::vector<WorldVector<double> > > *f,
		AbstractFunction<WorldVector<double>, 
		  WorldMatrix<double> > *divFct,
		bool symmetric);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo*, 
		     SubAssembler* ,
		     Quadrature *quad= NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    std::vector<DOFVectorBase<double>*> vecs_; 

    std::vector<DOFVectorBase<double>*> grads_;

    TertiaryAbstractFunction<WorldMatrix<double>, 
			     WorldVector<double>,
			     std::vector<double>, 
			     std::vector<WorldVector<double> > > *f_;

    AbstractFunction<WorldVector<double>, 
		     WorldMatrix<double> > *divFct_;

    WorldVector<double> *coordsAtQPs_;

    std::vector<double*> vecsAtQPs_;

    std::vector<WorldVector<double>*> gradsAtQPs_;

    bool symmetric_;
  };

  class GeneralParametric_SOT : public SecondOrderTerm
  {
  public:
    /// Constructor.
    GeneralParametric_SOT(std::vector<DOFVectorBase<double>*> vecs,
			  std::vector<DOFVectorBase<double>*> grads,
			  QuartAbstractFunction<WorldMatrix<double>, 
			  WorldVector<double>,
			  WorldVector<double>,
			  std::vector<double>, 
			  std::vector<WorldVector<double> > > *f,
			  AbstractFunction<WorldVector<double>, 
			  WorldMatrix<double> > *divFct,
			  bool symmetric);

    /** \brief
     * Implementation of \ref OperatorTerm::initElement().
     */
    void initElement(const ElInfo*, 
		     SubAssembler* ,
		     Quadrature *quad= NULL);

    /** \brief
     * Implements SecondOrderTerm::getLALt().
     */
    void getLALt(const ElInfo *elInfo, int nPoints, DimMat<double> **LALt) const;

    /** \brief
     * Implenetation of SecondOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *uhAtQP,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *D2UhAtQP,
	      double *result,
	      double factor) const;

    /** \brief
     * Implenetation of SecondOrderTerm::weakEval().
     */
    void weakEval(int nPoints,
		  const WorldVector<double> *grdUhAtQP,
		  WorldVector<double> *result) const;

  protected:
    std::vector<DOFVectorBase<double>*> vecs_; 

    std::vector<DOFVectorBase<double>*> grads_;

    QuartAbstractFunction<WorldMatrix<double>, 
			  WorldVector<double>,
			  WorldVector<double>,
			  std::vector<double>, 
			  std::vector<WorldVector<double> > > *f_;

    AbstractFunction<WorldVector<double>, 
		     WorldMatrix<double> > *divFct_;

    WorldVector<double> *coordsAtQPs_;
    WorldVector<double> elementNormal_;

    std::vector<double*> vecsAtQPs_;

    std::vector<WorldVector<double>*> gradsAtQPs_;

    bool symmetric_;
  };


  // ============================================================================
  // ===== class FirstOrderTerm =================================================
  // ============================================================================

  /**
   * \ingroup Assembler
   * 
   * \brief
   * Describes the first order terms: \f$ b \cdot \nabla u(\vec{x}) \f$
   */
  class FirstOrderTerm : public OperatorTerm
  {
  public:
    /// Constructor.
    FirstOrderTerm(int deg) 
      : OperatorTerm(deg) 
    {}

    /** \brief
     * Destructor.
     */
    virtual ~FirstOrderTerm() {}

    /** \brief
     * Evaluation of \f$ \Lambda b \f$.
     */
    virtual void getLb(const ElInfo *elInfo,
		       int nPoints, 
		       VectorOfFixVecs<DimVec<double> >& result) const = 0;

    /** \brief
     * Implenetation of FirstOrderTerm::eval().
     */
    void eval(int nPoints,
	      const double *,
	      const WorldVector<double> *grdUhAtQP,
	      const WorldMatrix<double> *,
	      double *result,
	      double factor) const 
    {
      int dow = Global::getGeo(WORLD);

      if (grdUhAtQP) {
	for (int iq = 0; iq < nPoints; iq++) {
	  double resultQP = 0.0;
	  for (int i = 0; i < dow; i++) {
	    resultQP += grdUhAtQP[iq][i];
	  }
	  result[iq] += resultQP * factor;
	}
      }