Over-determined formulation of the immersed boundary conditions method

A spectral fictitious domain method with internal forcing for solving elliptic PDEs

“…However, because the constraints are imposed on the values of the solution at points {P j } in the physical space, the matrices B and C are full matrices of size NP Â N c , that couple all the spectral components fû np g. Thus, the matrix in (22) is not sparse and its size (NP Â N c ) 2 precludes the use of a classical direct solver to calculate the solution of (22). Using the DLM within a spectral element framework, Dong et al [12] obtained a similar system and proposed to solve it iteratively by enforcing the constraints through an explicit penalty method.…”

Spectrally-accurate algorithm for the analysis of flows in two-dimensional vibrating channels

“…As the algorithm involves the use of two Fourier expansions, one for the field variables and one for the boundary conditions, it can be concluded that the convergence rate of the expansion representing the boundary conditions slows down as the geometry becomes more extreme. This suggests that the use of more Fourier modes in the expansion for the boundary conditions could increase the accuracy but adopting this method leads to an overdetermined formulation [37].…”

“…the so-called classical formulation [37], boundary relations corresponding to the lowest Fourier modes are used as the closing conditions; relations not used provide a measure of error in the enforcement of the flow boundary conditions as well as a test for the consistency of the method. Direct enforcement of a larger number of boundary relations leads to an overdetermined formulation of the IBC method which is advantageous in the case of more extreme geometries [37]. This formulation will be discussed in Section 9.…”

“…The method has been extended to two-dimensional unsteady problems [28], moving boundary problems involving Laplace [29] and biharmonic [30] operators, the complete Navier-Stokes system [31], to operators involving different classes of non-Newtonian fluids [32,33], to three-dimensional operators [34,35] as well as to operators expressed in cylindrical coordinate systems [36]. Its accuracy has been improved through the use of the overdetermined formulation [37]. The efficiency has been increased by an order of magnitude through the development of specialized solvers which account for the special structure of the coefficient matrix [38,39].…”

On the spectral accuracy of a fictitious domain method for elliptic operators in multi-dimensions

“…Another group of methods for the spectral approximation in complex domains are the algebraic fictitious domain methods, where the fictitious boundary conditions are imposed directly in the linear system of discretized equations. There are several variants of such an approach [7,[13][14][15]18,19] and they can be efficient, but typically they assume that the solution is analytic in the extended regular domain and often in real applications, this does not hold [3].…”