Fictitious domain method: A stabilized post-processing technique for boundary-flux calculation using cut elements

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This paper presents a technique for calculating boundary fluxes in the context of fictitious domain methods. We focus on a simple boundary-value problem and use Nitsche’s method stabilized with ghost penalty for the finite element formulation. To recover the flux, we derive a formulation directly from the boundary-value problem’s finite element formulation. Since the boundary may not align with the mesh, we compute the approximate flux using piecewise linear polynomials on cut elements. We then deduce the desired flux as the trace of the solution on the boundary. To ensure that the condition number of the resulting system matrix is independent of the boundary’s position on the mesh, we add a ghost penalty term. This term acts on the jumps of the gradients over interior facets belonging to elements intersected by the boundary. Two and three-dimensional numerical examples are provided, and show that the method is accurate and has optimal convergence regardless of the immersed boundary position.