An immersed interface method for Pennes bioheat transfer equation
Champike Attanayake So-Hsiang Chou
Discrete & Continuous Dynamical Systems - B 2015, 20(2): 323-337 doi: 10.3934/dcdsb.2015.20.323
We consider an immersed finite element method for solving one dimensional Pennes bioheat transfer equation with discontinuous coefficients and nonhomogenous flux jump condition. Convergence properties of the semidiscrete and fully discrete schemes are investigated in the $L^{2}$ and energy norms. By using the computed solution from the immerse finite element method, an inexpensive and effective flux recovery technique is employed to approximate flux over the whole domain. Optimal order convergence is proved for the immersed finite element approximation and its flux. Results of the simulation confirm the convergence analysis.
keywords: superconvergence immersed interface method immersed finite element method error analysis. Flux recovery
An immersed linear finite element method with interface flux capturing recovery
So-Hsiang Chou
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2343-2357 doi: 10.3934/dcdsb.2012.17.2343
A flux recovery technique is introduced for the computed solution of an immersed finite element method for one dimensional second-order elliptic problems. The recovery is by a cheap formula evaluation and is carried out over a single element at a time while ensuring the continuity of the flux across the interelement boundaries and the validity of the discrete conservation law at the element level. Optimal order rates are proved for both the primary variable and its flux. For piecewise constant coefficient problems our method can capture the flux at nodes and at the interface points exactly. Moreover, it has the property that errors in the flux are all the same at all nodes and interface points for general problems. We also show second order pressure error and first order flux error at the nodes. Numerical examples are provided to confirm the theory.
keywords: immersed interface method Flux recovery immersed finite element method superconvergence error analysis.

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