Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single-phase fluid in a porous medium in ޒ d , d Յ 3, subject to Forchhheimer's law-a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L ϱ ( J; L 2 (⍀)) and in L ϱ ( J; H(div; ⍀)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L ϱ ( J; L ϱ (⍀)) for the pressure.
A new image reconstruction algorithm is proposed to visualize static conductivity images of a subject in magnetic resonance electrical impedance tomography (MREIT). Injecting electrical current into the subject through surface electrodes, we can measure the induced internal magnetic flux density B = (Bx, By, Bz) using an MRI scanner. In this paper, we assume that only the z-component Bz is measurable due to a practical limitation of the measurement technique in MREIT. Under this circumstance, a constructive MREIT imaging technique called the harmonic Bz algorithm was recently developed to produce high-resolution conductivity images. The algorithm is based on the relation between inverted delta2Bz and the conductivity requiring the computation of inverted delta2Bz. Since twice differentiations of noisy Bz data tend to amplify the noise, the performance of the harmonic Bz algorithm is deteriorated when the signal-to-noise ratio in measured Bz data is not high enough. Therefore, it is highly desirable to develop a new algorithm reducing the number of differentiations. In this work, we propose the variational gradient Bz algorithm where Bz is differentiated only once. Numerical simulations with added random noise confirmed its ability to reconstruct static conductivity images in MREIT. We also found that it outperforms the harmonic Bz algorithm in terms of noise tolerance. From a careful analysis of the performance of the variational gradient Bz algorithm, we suggest several methods to further improve the image quality including a better choice of basis functions, regularization technique and multilevel approach. The proposed variational framework utilizing only Bz will lead to different versions of improved algorithms.
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