Analysis of FETI methods for multiscale PDEs TeX Export

@article{citeulike:3626668, abstract = {Abstract\ \ In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function \$\${\alpha \in L^\infty(\Omega)}\$\$ the condition number of the preconditioned FETI system can be bounded by C(α)\ (1\ + log(H/h))2 where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if \$\${\alpha|\_{\Omega\_{i}}}\$\$ varies only mildly in a layer Ω i,η of width η near the boundary of each of the subdomains Ω i , then \$\${C(\alpha) = \mathcal{O}((H/\eta)^2)}\$\$ , independent of the variation of α in the remainder Ω i \Ω i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests.}, author = {Pechstein, Clemens and Scheichl, Robert}, citeulike-article-id = {3626668}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/s00211-008-0186-2}, citeulike-linkout-1 = {http://www.springerlink.com/content/y643k346h2v867q4}, day = {1}, doi = {10.1007/s00211-008-0186-2}, journal = {Numerische Mathematik}, keywords = {debonding-microscale, fibre-reinforced-debonding}, month = {December}, number = {2}, pages = {293--333}, posted-at = {2008-11-21 15:35:42}, priority = {2}, title = {Analysis of FETI methods for multiscale PDEs}, url = {http://dx.doi.org/10.1007/s00211-008-0186-2}, volume = {111}, year = {2008} }

TY - JOUR ID - citeulike:3626668 L3 - citeulike-article-id:3626668 N2 - Abstract  In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function $${\alpha \in L^\infty(\Omega)}$$ the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))2 where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if $${\alpha|_{\Omega_{i}}}$$ varies only mildly in a layer Ω i,η of width η near the boundary of each of the subdomains Ω i , then $${C(\alpha) = \mathcal{O}((H/\eta)^2)}$$ , independent of the variation of α in the remainder Ω i \Ω i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests. IS - 2 JF - Numerische Mathematik EP - 333 TI - Analysis of FETI methods for multiscale PDEs VL - 111 SP - 293 KW - debonding-microscale KW - fibre-reinforced-debonding AU - Pechstein, Clemens AU - Scheichl, Robert PY - 2008/12/01/ UR - http://dx.doi.org/10.1007/s00211-008-0186-2 ER -