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

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 that of 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,then interpolate it, transform the interpolated quantity back into weak form and eventually assign it to the master side. Incase of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. The proposed methodis named INTERNODES (INTERpolation for NOnconforming DEcompositionS): it can be regarded as an alternative to the mortar element method and it is simpler to implement in a numerical code. We show on two dimensional problems that byusing 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.

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

Simone Deparis, Davide Forti, Alfio Quarteroni

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 that of 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, then interpolate it, transform the interpolated quantity back into weak form and eventually assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. The proposed method is named INTERNODES (INTERpolation for NOnconforming DEcompositionS): it can be regarded as an alternative to the mortar element method and it is simpler to implement in a numerical code. We show on two dimension al 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.

Elsevier2016

Numerical Approximation of Partial Differential Equations

Alfio Quarteroni

This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

Springer Berlin Heidelberg1994

Certified Reduced Basis Approximation for Parametrized Partial Differential Equations and Applications

Andrea Manzoni, Alfio Quarteroni, Gianluigi Rozza

Reduction strategies, such as model order reduction (MOR) or reduced basis (RB) methods, in scientic computing may become crucial in applications of increasing complexity. In this paper we review the reduced basis method (built upon a high-delity "truth" finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations, and comment on their potential impact on applications of industrial interest. The essential ingredients of RB methodology are: a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform a competitive Offline- Online splitting in the computational procedure, and a rigorous a posteriori error estimation used for both the basis selection and the certication of the solution. The combination of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identication). After a brief excursus on the methodology, we focus on linear elliptic and parabolic problems, discussing some extensions to more general classes of problems and several perspectives of the ongoing research. We present some results from applications dealing with heat and mass transfer, conduction-convection phenomena, and thermal treatments.

2011

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

Simone Deparis, Davide Forti, Alfio Quarteroni

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 that of 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, then interpolate it, transform the interpolated quantity back into weak form and eventually assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant. The proposed method is named INTERNODES (INTERpolation for NOnconforming DEcompositionS): it can be regarded as an alternative to the mortar element method and it is simpler to implement in a numerical code. We show on two dimension al 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.

Elsevier2016