Bilinear-forms

A bilinear form is defined by

define bilinearform <name> <flaglist>
integrator1
integrator2
integrator3
...

Example

define bilinearform a -fespace=v
laplace lam
robin alpha

A bilinear-form is always defined as sum over integrators. A bilinear-form maintains the stiffness matrix. For multi-level algorithms, a bilinear-form stores all matrices. Bilinear-forms for high order spaces have a bilinear-form for the corresponding lowest order space.

The following flags are defined

-fespace=<name>	bilinear form is defined on fe space <name>
-symmetric	bilinear form is symmetric (store just lower left triangular matrix)
-nonassemble	do not allocate matrix (bilinear-form is used, e.g., for post-processing)

An integrator is defined as

token <coef1> <coef2> ... <flaglist>

Example:

elasticity coef_e coef_nu -order=4

The <coefi> refers to a coefficient function defined above. It provides the coefficients defined sub-domain by dub-domain for integrators defined on the domain (e.g., laplace), or, the coefficient boundary-patch by boundary-patch for integrators defined on the surface (e.g., robin).

Allowed flags are

-order=num	use integration rule of order num. Default order is computed form element order.
-comp=num	use scalar integrator as component num for system (e.g., penalty term for y-displacement). num=0 adds integrator to all components.
-normal	add integrator in normal direction (penalty for normal-displacement)

The integrator tokens are

laplace lam	$\int_\Omega \lambda \nabla u \cdot \nabla v dx$
mass rho	$\int_\Omega \rho u v dx$
robin alpha	$\int_\Gamma \alpha u v ds$
elasticity e nu	$\int_\Omega D \varepsilon(u) : \varepsilon(v) dx$ (with $D$..3D elasticity tensor, or plane stress)
curlcurledge nu	$\int_\Omega \nu (\nabla \times u) (\nabla \times v) dx$ for $H(curl)$ spaces
massedge sigma	$\int_\Omega \sigma u \cdot v dx$ for $H(curl)$ spaces
robinedge sigma	$\int_\Gamma \sigma (n \times u) (n \times v) ds$ for $H(curl)$ spaces