volumential.function_extension

Function extension with regularity constraints.

This module owns the extension operators that continue a function given on a 2D curved domain to the surrounding box, together with the Goursat-type sumpy kernels the biharmonic extension is written in.

1. \(L^2\): extend with constant value

2. \(C^0\): harmonic extension

3. \(C^1\): biharmonic extension

volumential.function_extension.setup_cl_ctx(ctx=None, queue=None)

Return an OpenCL context, taken from ctx, queue, or created.

volumential.function_extension.setup_command_queue(ctx=None, queue=None)

Return a command queue, reusing queue when one is given.

volumential.function_extension.setup_array_context(actx=None, ctx=None, queue=None)

Return an array context, reusing actx when one is given.

volumential.function_extension.get_normal_vectors(queue, density_discr, loc_sign=-1)

Return the boundary normals pointing away from the extension domain.

Parameters:

loc_sign – -1 for the interior side (the discretization’s own normal), +1 for the exterior side (its negation).

volumential.function_extension.get_tangent_vectors(queue, density_discr, loc_sign)

Return the boundary tangents, oriented with the domain on the left.

volumential.function_extension.get_path_length(queue, density_discr)

Return the arclength of the boundary curve.

volumential.function_extension.get_arclength_parametrization_derivative(queue, density_discr, where=None)

Return x’(s), y’(s). s Follows the same direction as the parametrization. (Remark: t is the reference-to-global parametrization of the discretization).

volumential.function_extension.compute_constant_extension(queue, target_discr, qbx, density_discr, constant_val=0, actx=None)

Constant extension.

volumential.function_extension.compute_harmonic_extension(queue, target_discr, qbx, density_discr, f, loc_sign=1, target_association_tolerance=0.05, gmres_tolerance=1e-14, actx=None, representation_mode='auto')

Harmonic extension.

Parameters:

• loc_sign – Indicates the domain for extension, which equals the negation of the loc_sign for the original problem.

• representation_mode – One of “auto”, “s_plus_d”, “d_only”. “auto” uses D-only for interior extension (loc_sign=-1) and S+D otherwise.

class volumential.function_extension.ComplexLogKernel(dim=None)

Bases: ExpressionKernel

The Goursat kernel \(\log |z| / (4 \pi)\).

init_arg_names = ('dim',)

is_complex_valued = True

has_efficient_scale_adjustment = True

adjust_for_kernel_scaling(expr, rscale, nderivatives)

Efficient rescaling.

mapper_method = 'map_expression_kernel'

The name of the mapper method called for the kernel.

class volumential.function_extension.ComplexLinearLogKernel(dim=None)

Bases: ExpressionKernel

The Goursat kernel \(-\bar{z} \log |z| / (4 \pi)\).

init_arg_names = ('dim',)

is_complex_valued = True

has_efficient_scale_adjustment = False

mapper_method = 'map_expression_kernel'

The name of the mapper method called for the kernel.

class volumential.function_extension.ComplexLinearKernel(dim=None)

Bases: ExpressionKernel

The Goursat kernel \(-z / (8 \pi)\).

init_arg_names = ('dim',)

is_complex_valued = True

has_efficient_scale_adjustment = True

adjust_for_kernel_scaling(expr, rscale, nderivatives)

Efficient rescaling of the kernel.

mapper_method = 'map_expression_kernel'

The name of the mapper method called for the kernel.

class volumential.function_extension.ComplexFractionalKernel(dim=None)

Bases: ExpressionKernel

The Goursat kernel \(-i \bar{z} / (4 \pi z)\).

init_arg_names = ('dim',)

is_complex_valued = True

has_efficient_scale_adjustment = False

mapper_method = 'map_expression_kernel'

The name of the mapper method called for the kernel.

volumential.function_extension.get_extension_bie_symbolic_operator(loc_sign=1)

Return the symbolic Stokes BIE operator used by the extension.

Parameters:

loc_sign – -1 for interior Dirichlet, +1 for exterior Dirichlet.

volumential.function_extension.compute_biharmonic_extension(queue, target_discr, qbx, density_discr, f, fx, fy, target_association_tolerance=0.05, enforce_affine_match=False, actx=None, loc_sign=1)

Biharmoc extension. Currently only support 2D domains.

Parameters:

• loc_sign – Side for extension evaluation, following the same convention used by pytential layer potentials. +1 evaluates the exterior side and -1 evaluates the interior side.

• enforce_affine_match – If True, perform a boundary least-squares affine correction.