Macros
#define	LOBATTO_PHI0(x) (-2.0 * 1.22474487139158904909864203735)
	Definitions taken from Hermes2d code.

#define	LOBATTO_PHI1(x) (-2.0 * 1.58113883008418966599944677222 * (x))

#define	LOBATTO_PHI2(x) (-1.0 / 2.0 * 1.87082869338697069279187436616 * (5 * (x) * (x)-1))

#define	LOBATTO_PHI3(x) (-1.0 / 2.0 * 2.12132034355964257320253308631 * (7 * (x) * (x)-3) * (x))

#define	LOBATTO_PHI4(x)

#define	LOBATTO_PHI5(x)

#define	LOBATTO_PHI6(x)

#define	LOBATTO_PHI7(x)

#define	LOBATTO_PHI8(x)

#define	LOBATTO_PHI9(x)

#define	LOBATTO_PHI0X(x) (0)
	Derivatives of kernel functions for Lobbatto base.

#define	LOBATTO_PHI1X(x) (-2.0 * 1.58113883008418966599944677222)

#define	LOBATTO_PHI2X(x) (-1.0 / 2.0 * 1.87082869338697069279187436616 * (10 * (x)))

#define	LOBATTO_PHI3X(x) (-1.0 / 2.0 * 2.12132034355964257320253308631 * (21.0 * (x) * (x)-3.0))

#define	LOBATTO_PHI4X(x)

#define	LOBATTO_PHI5X(x)

#define	LOBATTO_PHI6X(x)

#define	LOBATTO_PHI7X(x)

#define	LOBATTO_PHI8X(x)

#define	LOBATTO_PHI9X(x)

Functions
PetscErrorCode	Legendre_polynomials (int p, double s, double diff_s, double L, double *diffL, const int dim)
	Calculate Legendre approximation basis.

PetscErrorCode	Jacobi_polynomials (int p, double alpha, double x, double t, double diff_x, double diff_t, double L, double diffL, const int dim)
	Calculate Jacobi approximation basis.

PetscErrorCode	IntegratedJacobi_polynomials (int p, double alpha, double x, double t, double diff_x, double diff_t, double L, double diffL, const int dim)
	Calculate integrated Jacobi approximation basis.

PetscErrorCode	Lobatto_polynomials (int p, double s, double diff_s, double L, double *diffL, const int dim)
	Calculate Lobatto base functions [24].

PetscErrorCode	LobattoKernel_polynomials (int p, double s, double diff_s, double L, double *diffL, const int dim)
	Calculate Kernel Lobatto base functions.

Macro Definition Documentation

◆ LOBATTO_PHI0

#define LOBATTO_PHI0 ( x ) (-2.0 * 1.22474487139158904909864203735)

Definitions taken from Hermes2d code.

kernel functions for Lobatto base

Definition at line 126 of file base_functions.h.

◆ LOBATTO_PHI0X

#define LOBATTO_PHI0X ( x ) (0)

Derivatives of kernel functions for Lobbatto base.

Definition at line 157 of file base_functions.h.

◆ LOBATTO_PHI1

#define LOBATTO_PHI1 ( x ) (-2.0 * 1.58113883008418966599944677222 * (x))

Definition at line 127 of file base_functions.h.

◆ LOBATTO_PHI1X

#define LOBATTO_PHI1X ( x ) (-2.0 * 1.58113883008418966599944677222)

Definition at line 158 of file base_functions.h.

◆ LOBATTO_PHI2

#define LOBATTO_PHI2 ( x ) (-1.0 / 2.0 * 1.87082869338697069279187436616 * (5 * (x) * (x)-1))

Definition at line 128 of file base_functions.h.

◆ LOBATTO_PHI2X

#define LOBATTO_PHI2X ( x ) (-1.0 / 2.0 * 1.87082869338697069279187436616 * (10 * (x)))

Definition at line 159 of file base_functions.h.

◆ LOBATTO_PHI3

#define LOBATTO_PHI3 ( x ) (-1.0 / 2.0 * 2.12132034355964257320253308631 * (7 * (x) * (x)-3) * (x))

Definition at line 130 of file base_functions.h.

◆ LOBATTO_PHI3X

#define LOBATTO_PHI3X ( x ) (-1.0 / 2.0 * 2.12132034355964257320253308631 * (21.0 * (x) * (x)-3.0))

Definition at line 161 of file base_functions.h.

◆ LOBATTO_PHI4

#define LOBATTO_PHI4 ( x )

Value:

(-1.0 / 4.0 * 2.34520787991171477728281505677 * \

(21 * (x) * (x) * (x) * (x)-14 * (x) * (x) + 1))

Definition at line 132 of file base_functions.h.

◆ LOBATTO_PHI4X

#define LOBATTO_PHI4X ( x )

Value:

(-1.0 / 4.0 * 2.34520787991171477728281505677 * \

((84.0 * (x) * (x)-28.0) * (x)))

Definition at line 163 of file base_functions.h.

◆ LOBATTO_PHI5

#define LOBATTO_PHI5 ( x )

Value:

(-1.0 / 4.0 * 2.54950975679639241501411205451 * \

((33 * (x) * (x)-30) * (x) * (x) + 5) * (x))

Definition at line 135 of file base_functions.h.

◆ LOBATTO_PHI5X

#define LOBATTO_PHI5X ( x )

Value:

(-1.0 / 4.0 * 2.54950975679639241501411205451 * \

((165.0 * (x) * (x)-90.0) * (x) * (x) + 5.0))

Definition at line 166 of file base_functions.h.

◆ LOBATTO_PHI6

#define LOBATTO_PHI6 ( x )

Value:

(-1.0 / 32.0 * 2.73861278752583056728484891400 * \

(((429 * (x) * (x)-495) * (x) * (x) + 135) * (x) * (x)-5))

Definition at line 138 of file base_functions.h.

◆ LOBATTO_PHI6X

#define LOBATTO_PHI6X ( x )

Value:

(-1.0 / 32.0 * 2.73861278752583056728484891400 * \

(((2574.0 * (x) * (x)-1980.0) * (x) * (x) + 270.0) * (x)))

Definition at line 169 of file base_functions.h.

◆ LOBATTO_PHI7

#define LOBATTO_PHI7 ( x )

Value:

(-1.0 / 32.0 * 2.91547594742265023543707643877 * \

(((715 * (x) * (x)-1001) * (x) * (x) + 385) * (x) * (x)-35) * (x))

Definition at line 141 of file base_functions.h.

◆ LOBATTO_PHI7X

#define LOBATTO_PHI7X ( x )

Value:

(-1.0 / 32.0 * 2.91547594742265023543707643877 * \

(((5005.0 * (x) * (x)-5005.0) * (x) * (x) + 1155.0) * (x) * (x)-35.0))

Definition at line 172 of file base_functions.h.

◆ LOBATTO_PHI8

#define LOBATTO_PHI8 ( x )

Value:

  (-1.0 / 64.0 * 3.08220700148448822512509619073 *                             \
   ((((2431 * (x) * (x)-4004) * (x) * (x) + 2002) * (x) * (x)-308) * (x) *     \
        (x) +                                                                  \
    7))

Definition at line 144 of file base_functions.h.

◆ LOBATTO_PHI8X

#define LOBATTO_PHI8X ( x )

Value:

  (-1.0 / 64.0 * 3.08220700148448822512509619073 *                             \
   ((((19448.0 * (x) * (x)-24024.0) * (x) * (x) + 8008.0) * (x) * (x)-616.0) * \
    (x)))

Definition at line 175 of file base_functions.h.

◆ LOBATTO_PHI9

#define LOBATTO_PHI9 ( x )

Value:

  (-1.0 / 128.0 * 6.4807406984078603784382721642 *                             \
   ((((4199 * (x) * (x)-7956) * (x) * (x) + 4914) * (x) * (x)-1092) * (x) *    \
        (x) +                                                                  \
    63) *                                                                      \
   (x))

Definition at line 149 of file base_functions.h.

◆ LOBATTO_PHI9X

#define LOBATTO_PHI9X ( x )

Value:

  (-1.0 / 128.0 * 6.4807406984078603784382721642 *                             \
   ((((37791.0 * (x) * (x)-55692.0) * (x) * (x) + 24570.0) * (x) *             \
     (x)-3276.0) *                                                             \
    (x) * (x)-63.0))

Definition at line 179 of file base_functions.h.

Function Documentation

◆ IntegratedJacobi_polynomials()

PetscErrorCode IntegratedJacobi_polynomials	(	int	p,
		double	alpha,
		double	x,
		double	t,
		double *	diff_x,
		double *	diff_t,
		double *	L,
		double *	diffL,
		const int	dim
	)

Calculate integrated Jacobi approximation basis.

For more details see [25]

Parameters

p	is approximation order
alpha	polynomial parameter
x	is position \(s\in[0,t]\)
t	range of polynomial
diff_x	derivatives of shape functions, i.e. \(\frac{\partial x}{\partial \xi_i}\)
diff_t	derivatives of shape functions, i.e. \(\frac{\partial t}{\partial \xi_i}\)

Return values

L	approximation functions
diffL	derivatives, i.e. \(\frac{\partial L}{\partial \xi_i}\)

Parameters

dim dimension

Returns: error code

Definition at line 134 of file base_functions.c.

                                                                          {
  MoFEMFunctionBeginHot;
#ifndef NDEBUG
  if (dim < 1)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
  if (dim > 3)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
  if (p < 1)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 1");
#endif // NDEBUG
  L[0] = x;
  if (diffL != NULL) {
    int d = 0;
    for (; d != dim; ++d) {
      diffL[d * p + 0] = diff_x[d];
    }
  }
  if (p == 0)
    MoFEMFunctionReturnHot(0);
  double jacobi[(p + 1)];
  double diff_jacobi[(p + 1) * dim];
  ierr = Jacobi_polynomials(p, alpha, x, t, diff_x, diff_t, jacobi, diff_jacobi,
                            dim);
  CHKERRQ(ierr);
  int l = 1;
  for (; l < p; l++) {
    int i = l + 1;
    const double a = (i + alpha) / ((2 * i + alpha - 1) * (2 * i + alpha));
    const double b = alpha / ((2 * i + alpha - 2) * (2 * i + alpha));
    const double c = (i - 1) / ((2 * i + alpha - 2) * (2 * i + alpha - 1));
    L[l] = a * jacobi[i] + b * t * jacobi[i - 1] - c * t * t * jacobi[i - 2];
    if (diffL != NULL) {
      int d = 0;
      for (; d != dim; ++d) {
        diffL[d * p + l] = a * diff_jacobi[d * (p + 1) + i] +
                            b * (t * diff_jacobi[d * (p + 1) + i - 1] +
                                 diff_t[d] * jacobi[i - 1]) -
                            c * (t * t * diff_jacobi[d * (p + 1) + i - 2] +
                                 2 * t * diff_t[d] * jacobi[i - 2]);
      }
    }
  }
  MoFEMFunctionReturnHot(0);
}

◆ Jacobi_polynomials()

PetscErrorCode Jacobi_polynomials	(	int	p,
		double	alpha,
		double	x,
		double	t,
		double *	diff_x,
		double *	diff_t,
		double *	L,
		double *	diffL,
		const int	dim
	)

Calculate Jacobi approximation basis.

For more details see [25]

Parameters

p	is approximation order
alpha	polynomial parameter
x	is position \(s\in[0,t]\)
t	range of polynomial
diff_x	derivatives of shape functions, i.e. \(\frac{\partial x}{\partial \xi_i}\)
diff_t	derivatives of shape functions, i.e. \(\frac{\partial t}{\partial \xi_i}\)

Return values

L	approximation functions
diffL	derivatives, i.e. \(\frac{\partial L}{\partial \xi_i}\)

Parameters

dim dimension

Returns: error code

Definition at line 67 of file base_functions.c.

                                                                {
  MoFEMFunctionBeginHot;
#ifndef NDEBUG
  if (dim < 1)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
  if (dim > 3)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
  if (p < 0)
    SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 0");
 
  if (diffL != NULL) {
    if (diff_x == NULL) {
      SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "diff_s == NULL");
    }
  }
 
#endif // NDEBUG
  L[0] = 1;
  if (diffL != NULL) {
    diffL[0 * (p + 1) + 0] = 0;
    if (dim >= 2) {
      diffL[1 * (p + 1) + 0] = 0;
      if (dim == 3) {
        diffL[2 * (p + 1) + 0] = 0;
      }
    }
  }
  if (p == 0)
    MoFEMFunctionReturnHot(0);
  L[1] = 2 * x - t + alpha * x;
  if (diffL != NULL) {
    int d = 0;
    for (; d < dim; ++d) {
      double d_t = (diff_t) ? diff_t[d] : 0;
      diffL[d * (p + 1) + 1] = (2 + alpha) * diff_x[d] - d_t;
    }
  }
  if (p == 1)
    MoFEMFunctionReturnHot(0);
  int l = 1;
  for (; l < p; l++) {
    int lp1 = l + 1;
    double a = 2 * lp1 * (lp1 + alpha) * (2 * lp1 + alpha - 2);
    double b = 2 * lp1 + alpha - 1;
    double c = (2 * lp1 + alpha) * (2 * lp1 + alpha - 2);
    double d = 2 * (lp1 + alpha - 1) * (lp1 - 1) * (2 * lp1 + alpha);
    double A = b * (c * (2 * x - t) + alpha * alpha * t) / a;
    double B = d * t * t / a;
    L[lp1] = A * L[l] - B * L[l - 1];
    if (diffL != NULL) {
      int z = 0;
      for (; z < dim; ++z) {
        double d_t = (diff_t) ? diff_t[z] : 0;
        double diffA =
            b * (c * (2 * diff_x[z] - d_t) + alpha * alpha * d_t) / a;
        double diffB = d * 2 * t * d_t / a;
        diffL[z * (p + 1) + lp1] = A * diffL[z * (p + 1) + l] -
                                   B * diffL[z * (p + 1) + l - 1] +
                                   diffA * L[l] - diffB * L[l - 1];
      }
    }
  }
  MoFEMFunctionReturnHot(0);
}

Macros

Functions

Macro Definition Documentation

◆ LOBATTO_PHI0

◆ LOBATTO_PHI0X

◆ LOBATTO_PHI1

◆ LOBATTO_PHI1X

◆ LOBATTO_PHI2

◆ LOBATTO_PHI2X

◆ LOBATTO_PHI3

◆ LOBATTO_PHI3X

◆ LOBATTO_PHI4

◆ LOBATTO_PHI4X

◆ LOBATTO_PHI5

◆ LOBATTO_PHI5X

◆ LOBATTO_PHI6

◆ LOBATTO_PHI6X

◆ LOBATTO_PHI7

◆ LOBATTO_PHI7X

◆ LOBATTO_PHI8

◆ LOBATTO_PHI8X

◆ LOBATTO_PHI9

◆ LOBATTO_PHI9X

Function Documentation

◆ IntegratedJacobi_polynomials()

◆ Jacobi_polynomials()