INTERNODES: an accurate interpolation-based method for coupling the Galerkin solutions of PDEs on subdomains featuring non-conforming interfaces

43/2015

INTERNODES: an accurate interpolation-based method for coupling the Galerkin solutions of PDEs on subdomains featuring non-conforming interfaces

Wednesday 9th September 2015

Deparis, S.; Forti, D.; Gervasio, P.; Quarteroni, A.

We are interested in the approximation of partial differential equations on domains decomposed into two (or several) subdomains featuring non-conforming interfaces. The non-conformity may be due to different meshes and/or different polynomial degrees used from the two sides, or even to a geometrical mismatch. Across each interface, one subdomain is identified as master and the other as slave. We consider Galerkin methods for the discretization (such as finite element or spectral element methods) that make use of two interpolants for transferring information across the interface: one from master to slave and another one from slave to master. The former is used to ensure continuity of the primal variable (the problem solution), while the latter for the dual variable (the normal flux). In particular, since the dual variable is expressed in weak form, we first compute a strong representation of the dual variable from the slave side, interpolate it, transform the interpolated quantity back into weak form and assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. We name the proposed method INTERNODES (INTERpolation for NOnconforming DEcompositionS). It can be regarded as an alternative to the mortar element method and it is much simpler to implement in a numerical code. We show on two dimensional problems that by using the Lagrange interpolation we obtain at least as good convergence results as with the mortar element method with any order of polynomials. When using low order polynomials, the radial-basis interpolant leads to the same convergence properties as the Lagrange interpolant. We conclude with a comparison between INTERNODES and a standard conforming approximation in a three dimensional case.