Interface problems have wide applications in modern scientific research. Obtaining accurate numerical solutions of multi-domain problems involving triple junction conditions remains a significant challenge. In this paper, we develop an efficient finite element method based on non-body-fitting meshes for solving multi-domain elliptic interface problems. We follow the idea of immersed finite element by modifying local basis functions to accommodate interface conditions. We enrich the local finite element space by adding new basis functions for handling non-homogeneous flux jump. The numerical scheme is symmetric and positive definite. Numerical experiments are provided to demonstrate the features of our method.
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Additive models for symmetric positive-definite matrices and Lie groups
We propose and investigate an additive regression model for symmetric positive-definite matrix-valued responses and multiple scalar predictors. The model exploits the Abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general Abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions, but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established, and numerical studies show that the proposed model enjoys good numerical performance, and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.
more » « less- Award ID(s):
- 2014626
- PAR ID:
- 10413350
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 110
- Issue:
- 2
- ISSN:
- 0006-3444
- Page Range / eLocation ID:
- p. 361-379
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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