DOF Constraints

CutFEMx solves PDEs using finite element spaces defined on a background mesh. Only part of that mesh may intersect the physical domain. Degrees of freedom whose basis functions do not touch the active domain must be constrained before solving; otherwise the assembled matrix can contain zero rows or uncontrolled nullspace components.

Active and inactive degrees of freedom

Let \(V_h=\operatorname{span}\{\varphi_i\}\) be the finite element space on the background mesh. For a problem posed on \(\Omega_h\), the physically active index set is

\[ {\mathcal I}_\Omega = \{i:\operatorname{supp}(\varphi_i)\cap\Omega_h\ne\emptyset\}. \]

An index \(i\notin{\mathcal I}_\Omega\) has no physical volume, boundary, or stabilization contribution. The corresponding row in the assembled system is therefore not determined by the PDE unless it is constrained algebraically.

Active Domain From A Form

After compiling a CutFEMx form, derive the active domain from that form:

a_form = cutfemx.fem.form(a)
L_form = cutfemx.fem.form(L)

active_domain = cutfemx.fem.active_domain(a_form)

The active-domain object is derived from the integration domains of the form. This is important: the active degrees of freedom are not determined by the level set alone, but by the measures that actually appear in the assembled problem.

Form-dependent activity

If a form contains integrals only over \(\Omega_h\), then activity is determined by basis functions with support on \(\Omega_h\). If additional skeleton or stabilization terms are present, the active support may include neighbouring cells touched by those terms. The active-domain query therefore belongs after form construction, not before it.

Deactivation

For MatrixCSR assembly, assemble and deactivate outside the active domain:

b = cutfemx.fem.assemble_vector(L_form)
A = cutfemx.fem.assemble_matrix(a_form)

cutfemx.fem.deactivate_outside(A, b, active_domain)

For PETSc assembly, use the PETSc helpers:

import cutfemx.petsc

A = cutfemx.petsc.assemble_matrix(a_form)
b = cutfemx.petsc.assemble_vector(L_form)

cutfemx.petsc.deactivate_outside(A, b, active_domain)

What deactivation does

For an inactive degree of freedom \(i\), deactivation replaces the row by a simple identity constraint,

\[ A_{ii}=1,\qquad A_{ij}=0\quad(j\ne i),\qquad b_i=0. \]

This keeps the linear system nonsingular while leaving the active finite element problem unchanged. Active rows are not modified by this operation.

Block Systems

Interface and multi-domain problems often use one finite element space per phase. The active set is then block dependent. For a two-phase problem with unknowns \((u_1,u_2)\), the algebraic system has the form

\[\begin{split} \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} U_1\\ U_2 \end{bmatrix} = \begin{bmatrix} B_1\\ B_2 \end{bmatrix}. \end{split}\]

The inactive degrees of freedom for \(u_1\) and \(u_2\) are generally different, because \(\Omega_1\) and \(\Omega_2\) occupy different parts of the background mesh. Use block deactivation for such systems:

domain1 = cutfemx.fem.active_domain(a_forms[0][0])
domain2 = cutfemx.fem.active_domain(a_forms[1][1])

cutfemx.fem.deactivate_outside_blocks(
    A_matrix_blocks,
    [domain1, domain2],
    b_vector_blocks,
)

The interface Poisson example uses this pattern.

Diagnostics

The active-domain object exposes arrays that are useful for debugging:

active_cells = active_domain.active_cells
inactive_dofs = active_domain.inactive_dofs
indicator = active_domain.indicator

indicator is a DOLFINx function that can be visualized on the background mesh. It is often the fastest way to detect a selector error: if the active region is empty, shifted, or inverted, the inactive-DOF pattern will show it immediately.

After deactivation, zero-row diagnostics can be used as a final algebraic check:

zero_rows = cutfemx.fem.zero_rows(A)

For a well-constrained scalar problem, no zero rows should remain after deactivation. In block systems, use the corresponding block-row diagnostic.