<abstract xmlns="http://www.w3.org/1999/xhtml"><p>New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of <xref ref-type="bibr" rid="j_amcs-2017-0037_ref_027_w2aab3b7b6b1b6b1ab1ac27Aa">Wheeler (1974)</xref>. For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.</p></abstract>

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution, but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea of Wheeler (1974). For 2D interface problems, the point is to introduce a small tube near the interface and propose the gradient as part of unknowns, which is similar to a mixed finite element method, but only at the interface. Thus the computational cost is just slightly higher than in the standard finite element method. We present a rigorous one dimensional analysis, which shows a second order convergence order for both the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface.

Accurate gradient computations at interfaces using finite element methods

Nanjing University of Posts and Telecommunications

China Institute of Water Resource and Hydropower Research (IWHR)

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Accurate gradient computations at interfaces using finite element methods"}

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is to get not only an accurate solution,...

Accurate gradient computations at interfaces using finite element methods

Published Online: Sep 23, 2017

Page range: 527 - 537

Received: Nov 09, 2016

Accepted: May 31, 2017

DOI: https://doi.org/10.1515/amcs-2017-0037

Keywords
elliptic interface problems, gradient/flux computation, IFEM, mixed FE formulation, computational tube

© 2017 Fangfang Qin et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Accurate gradient computations at interfaces using finite element methods

Published Online: Sep 23, 2017

Page range: 527 - 537

Received: Nov 09, 2016

Accepted: May 31, 2017

DOI: https://doi.org/10.1515/amcs-2017-0037

Keywordselliptic interface problems, gradient/flux computation, IFEM, mixed FE formulation, computational tube

© 2017 Fangfang Qin et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
elliptic interface problems, gradient/flux computation, IFEM, mixed FE formulation, computational tube