Reduced Basis Methods with local mass conservation

Published version (open access)

Key ideas

• The difference between two locally conservative flux fields is solenoidal and therefore given by the curl of a potential field.
• We propose a three-step solution procedure:
  1. Given $f$, use a locally conservative scheme such as the TPFA finite volume method to find a $q_f$ that solves the mass balance equation: $\nabla \cdot q_f = f$.
  2. Solve for the potential $r$ by solving a curl-curl problem.
  3. Set $q = q_f + \nabla \times r$ and post-process the pressure $p$.
• The second step can be approximated using a Reduced Basis Method.

Three test cases solved using the reduced basis method. In each case, mass is conserved locally up to machine precision.

Main findings

• The solution is locally conservative regardless of the accuracy of the reduced basis method. This can be shown as follows: \(\nabla \cdot q = \nabla \cdot q_f + \nabla \cdot (\nabla \times r) = \nabla \cdot q_f = f.\)
• The original saddle point problem is reduced to three smaller, symmetric positive definite systems.
• The procedure is valid for mixed-dimensional fracture flow by using the mixed-dimensional curl operator.