puncturedfem.LocalPoissonFunction

Function with a polynomial Laplacian and continuous trace on the boundary.

Typically, this class is used to represent functions in the local Poisson space V_p(K), which consists of functions whose Laplacian is a polynomial of degree <= p-2 on each edge of K, and whose continuous trace on any edge of the boundary of K is the restriction of a polynomial to that edge. However, it can also be used to represent functions with an arbitrary continuous trace on the boundary of K. Any such function v can be decomposed as v = P + phi where P is a polynomial of degree <= p in K and phi is a harmonic function. (Note that this decomposition is not unique, since there are harmonic polynomials). We refer to P as the "polynomial part" and phi as the "harmonic part". Furthermore, in multiply connected cells the harmonic part can be decomposed as phi = psi + sum_{j=1}^m a_j log |x - xi_j| where xi_j is a fixed arbitrary point in the jth hole of K, and psi is a harmonic function with a harmonic conjugate psi_hat. We refer to psi as the "conjugable part". Given a parametrization x(tau) of the boundary of K, we refer to (d / d tau) v(x(tau)) = nabla v(x(tau)) * x'(tau) as the "weighted normal derivative" of v, which we commonly abbreviate in code with "wnd". Similarly, the "weighted tangential derivative" is abbreviated "wtd". An "anti-Laplacian" of u is any function U such that Delta U = u where Delta is the Laplacian operator. In particular, we desire an anti- Laplacian of the harmonic part phi, as well as its weighted normal derivative. This allows us to compute the L^2 inner product of two local functions v and w using only boundary integrals. Interior values of a local function v, as well as its gradient, can be computed using Cauchy's integral formula.

Attributes:

Name	Type	Description
harm	LocalHarmonic	The harmonic part, phi.
poly	LocalPolynomial	The polynomial part, P.
mesh_cell	MeshCell	The mesh cell on which the local function is defined.
int_vals	np.ndarray	Interior values of the local function, evaluated on the interior mesh defined by the mesh cell.
int_grad1	np.ndarray	First component of the gradient of the local function, evaluated on the interior mesh defined by the mesh cell.
int_grad2	np.ndarray	Second component of the gradient of the local function, evaluated on the interior mesh defined by the mesh cell.
key	GlobalKey	A unique tag that identifies the local function in the global space.

__add__(other)

Add two local Poisson functions.

Parameters:

Name	Type	Description	Default
other	LocalPoissonFunction	The other local Poisson function.	required

Returns:

Type	Description
LocalPoissonFunction	The sum of the two local Poisson functions.

__init__(nyst, laplacian=Polynomial(), trace=0, evaluate_interior=True, evaluate_gradient=False, key=None)

Build an element of the local Poisson space V_p(K).

Parameters:

Name	Type	Description	Default
nyst	NystromSolver	Nystrom solver object for solving integral equations.	required
laplacian	Polynomial, optional	Polynomial Laplacian of the local function, by default Polynomial().	Polynomial()
trace	DirichletTrace or FloatLike, optional	Dirichlet trace of the local function, by default 0. If a numpy array, must be the same length as the number of sampled points on the boundary.	0
evaluate_interior	bool, optional	Whether or not to compute the interior values, by default False.	True
evaluate_gradient	bool, optional	Whether or not to compute the gradient, by default False. Takes precedence over evaluate_interior.	False
key	Optional[GlobalKey], optional	A unique tag that identifies the local function in the global space.	None

__mul__(other)

Multiply the local Poisson function by a scalar.

Parameters:

Name	Type	Description	Default
other	Union[int, float]	The scalar.	required

Returns:

Type	Description
LocalPoissonFunction	The product of the local Poisson function and the scalar.

__rmul__(other)

Multiply the local Poisson function by a scalar.

Parameters:

Name	Type	Description	Default
other	Union[int, float]	The scalar.	required

Returns:

Type	Description
LocalPoissonFunction	The product of the local Poisson function and the scalar.

__sub__(other)

Subtract two local Poisson functions.

Parameters:

Name	Type	Description	Default
other	LocalPoissonFunction	The other local Poisson function.	required

Returns:

Type	Description
LocalPoissonFunction	The difference of the two local Poisson functions.

__truediv__(other)

Divide the local Poisson function by a scalar.

Parameters:

Name	Type	Description	Default
other	Union[int, float]	The scalar.	required

Returns:

Type	Description
LocalPoissonFunction	The division of the local Poisson function by the scalar.

compute_interior_values(compute_int_grad=True)

Compute the interior values.

Also compute the components of the gradient if compute_int_grad is True. The interior values are stored in self.int_vals, and the gradient components are stored in self.int_grad1 and self.int_grad2.

Parameters:

Name	Type	Description	Default
compute_int_grad	bool, optional	Whether or not to compute the gradient, by default True.	True

get_h1_semi_inner_prod(other)

Return the H^1 semi-inner product int_K grad(self) * grad(other) dx.

Parameters:

Name	Type	Description	Default
other	LocalPoissonFunction, LocalHarmonic, or LocalPolynomial	Another local function on the same mesh cell.	required

get_l2_inner_prod(other)

Return the L^2 inner product int_K self * other dx.

Parameters:

Name	Type	Description	Default
other	LocalPoissonFunction, LocalHarmonic, or LocalPolynomial	Another local function on the same mesh cell.	required

get_trace_values()

Return the trace values along the boundary of the mesh cell.

Returns:

Name	Type	Description
vals	numpy.ndarray	Values of the trace, traverse in the same order as the sampled points on the boundary of the mesh cell.