trace_matrices.py

Here we compute the trace matrices \mathbb{T}_{\text{N}}, \mathbb{T}_{\text{E}} and \mathbb{T}_{\text{F}}.

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trace_matrices.TE(N_xi, N_et, N_sg)[source]

The trace matrix \mathbb{T}_{\text{E}} for mimetic polynomials constructed on three sets of nodes, \left\lbrace\xi_0,\xi_1,\cdots, \xi_{N0} \right\rbrace, \left\lbrace\eta_0,\eta_1,\cdots, \eta_{N1}\right\rbrace and \left\lbrace\varsigma_0, \varsigma_1,\cdots, \varsigma_{N2}\right\rbrace.

Parameters:

  • N_xi (positive integer) – N_xi + 1 nodes in set \left\lbrace\xi_0, \xi_1,\cdots, \xi_{N_\xi} \right\rbrace.

  • N_et (positive integer) – N_et + 1 nodes in set \left\lbrace\eta_0, \eta_1,\cdots, \eta_{N_\eta} \right\rbrace.

  • N_sg (positive integer) – N_sg + 1 nodes in set \left\lbrace \varsigma_0, \varsigma_1,\cdots, \varsigma_{N_\varsigma} \right\rbrace.

Returns:

A csc_matrix \mathbb{T}_{\text{E}}.

Example:

trace_matrices.TF(N_xi, N_et, N_sg)[source]

The trace matrix \mathbb{T}_{\text{F}} for mimetic polynomials constructed on three sets of nodes, \left\lbrace\xi_0,\xi_1,\cdots, \xi_{N0} \right\rbrace, \left\lbrace\eta_0,\eta_1,\cdots, \eta_{N1}\right\rbrace and \left\lbrace\varsigma_0, \varsigma_1,\cdots, \varsigma_{N2}\right\rbrace.

Parameters:

  • N_xi (positive integer) – N_xi + 1 nodes in set \left\lbrace\xi_0, \xi_1,\cdots, \xi_{N_\xi} \right\rbrace.

  • N_et (positive integer) – N_et + 1 nodes in set \left\lbrace\eta_0, \eta_1,\cdots, \eta_{N_\eta} \right\rbrace.

  • N_sg (positive integer) – N_sg + 1 nodes in set \left\lbrace \varsigma_0, \varsigma_1,\cdots, \varsigma_{N_\varsigma} \right\rbrace.

Returns:

A csr_matrix \mathbb{T}_{\text{F}}.

Example:

>>> T = TF(2, 2, 2)
>>> T.indices
array([ 0,  3,  6,  9,  2,  5,  8, 11, 12, 13, 18, 19, 16, 17, 22, 23, 24,
       25, 26, 27, 32, 33, 34, 35], dtype=int32)
>>> T.data
array([-1, -1, -1, -1,  1,  1,  1,  1, -1, -1, -1, -1,  1,  1,  1,  1, -1,
       -1, -1, -1,  1,  1,  1,  1], dtype=int32)
trace_matrices.TN(N_xi, N_et, N_sg)[source]

The trace matrix \mathbb{T}_{\text{N}} for mimetic polynomials constructed on three sets of nodes, \left\lbrace\xi_0,\xi_1,\cdots, \xi_{N0} \right\rbrace, \left\lbrace\eta_0,\eta_1,\cdots, \eta_{N1}\right\rbrace and \left\lbrace\varsigma_0, \varsigma_1,\cdots, \varsigma_{N2}\right\rbrace.

Parameters:

  • N_xi (positive integer) – N_xi + 1 nodes in set \left\lbrace\xi_0, \xi_1,\cdots, \xi_{N_\xi} \right\rbrace.

  • N_et (positive integer) – N_et + 1 nodes in set \left\lbrace\eta_0, \eta_1,\cdots, \eta_{N_\eta} \right\rbrace.

  • N_sg (positive integer) – N_sg + 1 nodes in set \left\lbrace \varsigma_0, \varsigma_1,\cdots, \varsigma_{N_\varsigma} \right\rbrace.

Returns:

A csr_matrix \mathbb{T}_{\text{N}}.

Example:

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