Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

Abstract

This work focuses on the development of a non-conforming domain decomposition method for the approximation of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a discrete number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which also include the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains; one of the possible choices is to approximate the interface variational space by Fourier basis functions. As we show in the numerical simulations, our approach is well-suited for the solution of problems with non-conforming meshes or with finite element basis functions with different polynomial degrees in each subdomain. Another application of the method that still needs to be investigated is the coupling of solutions obtained from otherwise incompatible methods, such as the finite element method, the spectral element method or isogeometric analysis.

Model order reduction of flow based on a modular geometrical approximation of blood vessels

“…If ( u , p ) satisfies W1 , then ( , , { σ (u, p )n jm } m ∈ N ( j ) ) is also solution of the local problem W2 , for every j = 1, …, N Ω . The Lagrange multipliers therefore play the role of the stress at the interfaces; for details, see, e.g., [ 25 , 43 ]. Conversely, if ( u j , p j , { λ [ jm ] } m ∈ N ( j ) ) are the local solutions of W2 , then is solution of W1 .…”

“…The most popular approaches rely on the introduction of suitable Lagrange multipliers at the interfaces to enforce transmission conditions, as in W2 ; see, e.g., the well-known mortar method [ 44 – 46 ] and INTERNODES [ 47 , 48 ]. In this paper, we adopt the algorithm presented in [ 25 ], which is based on the discretization of the Lagrange multipliers space via a small number of spectral basis functions defined on the interfaces. For our application, this choice is convenient because (i) the method allows us to recover the h -convergence order of the primal discretization method—i.e., the FE method—even when a small number of spectral basis functions is considered, (ii) defining a spectral basis on each interface does not require to project nor to interpolate the traces of FE basis functions from one side to the other, and (iii) as already mentioned, the interfaces are circular in the target configuration, which allows us to employ a set of standard orthonormal basis functions on the two-dimensional disk.…”

“…For our application, this choice is convenient because (i) the method allows us to recover the h -convergence order of the primal discretization method—i.e., the FE method—even when a small number of spectral basis functions is considered, (ii) defining a spectral basis on each interface does not require to project nor to interpolate the traces of FE basis functions from one side to the other, and (iii) as already mentioned, the interfaces are circular in the target configuration, which allows us to employ a set of standard orthonormal basis functions on the two-dimensional disk. In the remainder of this section and in Section 4.2 we recall the basics of this nonconforming method applied to the Navier–Stokes equations; the interested reader is referred to [ 25 ] for the details.…”

“…The global flow approximations by our ROM is computed as a composition of local (to the subdomains) solutions, namely linear combinations of the basis functions defined in every subdomain scaled by the divergence-free Piola transformation. The local solutions are coupled by a nonconforming domain-decomposition method based on the use of spectral Lagrange multipliers on the 2D interfaces [ 25 ].…”

Model order reduction of flow based on a modular geometrical approximation of blood vessels

“…The global flow approximations by our ROM is computed as composition of local (to the subdomains) solutions, namely linear combinations of the basis functions defined in every subdomain scaled by the divergence-free Piola transformation. The local solutions are coupled by a nonconforming domain-decomposition method based on the use of spectral Lagrange multipliers on the 2D interfaces [22].…”