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| 1 | +"""Arbogast-Correa elements on quadrilaterals. |
| 2 | +
|
| 3 | +This element's definition appears in https://epubs.siam.org/doi/10.1137/15M1013705 |
| 4 | +(Arbogast, Correa, 2016) |
| 5 | +""" |
| 6 | + |
| 7 | +from ..finite_element import CiarletElement |
| 8 | +from ..moments import make_integral_moment_dofs |
| 9 | +from ..polynomials import polynomial_set, Hdiv_serendipity |
| 10 | +from ..functionals import NormalIntegralMoment, IntegralAgainst |
| 11 | +from .dpc import DPC |
| 12 | +from ..symbolic import x |
| 13 | + |
| 14 | + |
| 15 | +class AC(CiarletElement): |
| 16 | + """Arbogast-Correa Hdiv finite element.""" |
| 17 | + |
| 18 | + def __init__(self, reference, order, variant="equispaced"): |
| 19 | + poly = polynomial_set(reference.tdim, reference.tdim, order) |
| 20 | + if order == 0: |
| 21 | + poly += [(x[0], 0), (0, x[1])] |
| 22 | + else: |
| 23 | + poly += Hdiv_serendipity(reference.tdim, reference.tdim, order) |
| 24 | + poly += [(x[0] ** (i + 1) * x[1] ** (order - i), x[0] ** i * x[1] ** (1 + order - i)) |
| 25 | + for i in range(order + 1)] |
| 26 | + |
| 27 | + dofs = make_integral_moment_dofs( |
| 28 | + reference, |
| 29 | + facets=(NormalIntegralMoment, DPC, order, {"variant": variant}), |
| 30 | + ) |
| 31 | + |
| 32 | + for i in range(order + 1): |
| 33 | + for j in range(order + 1 - i): |
| 34 | + if i + j > 0: |
| 35 | + f = (i * x[0] ** (i - 1) * x[1] ** j, j * x[0] ** i * x[1] ** (j - 1)) |
| 36 | + dofs.append(IntegralAgainst(reference, f, entity=(2, 0), |
| 37 | + mapping="contravariant")) |
| 38 | + |
| 39 | + for i in range(1, order - 1): |
| 40 | + for j in range(1, order - i): |
| 41 | + f = ( |
| 42 | + x[0] ** i * (1 - x[0]) * x[1] ** (j - 1) * (j * (1 - x[1]) - x[1]), |
| 43 | + -x[0] ** (i - 1) * (i * (1 - x[0]) - x[0]) * x[1] ** j * (1 - x[1]) |
| 44 | + ) |
| 45 | + dofs.append(IntegralAgainst(reference, f, entity=(2, 0), |
| 46 | + mapping="contravariant")) |
| 47 | + |
| 48 | + super().__init__(reference, order, poly, dofs, reference.tdim, reference.tdim) |
| 49 | + self.variant = variant |
| 50 | + |
| 51 | + def init_kwargs(self): |
| 52 | + """Return the kwargs used to create this element.""" |
| 53 | + return {"variant": self.variant} |
| 54 | + |
| 55 | + names = ["Arbogast-Correa", "AC", "AC full", "Arbogast-Correa full"] |
| 56 | + references = ["quadrilateral"] |
| 57 | + min_order = 0 |
| 58 | + continuity = "H(div)" |
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