Abstract.A P 1 -nonconforming quadrilateral finite element is introduced for second-order elliptic problems in two dimensions. Unlike the usual quadrilateral nonconforming finite elements, which contain quadratic polynomials or polynomials of degree greater than 2, our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges. One of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements. An optimal rate of convergence is obtained. Also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral. An extension of the P 1 -nonconforming element to three dimensions is also given. Finally, several numerical results are reported to confirm the effective nature of the proposed new element. Key words. nonconforming finite elements, quadrilateral, elliptic problems AMS subject classifications. 65N30, 65N12, 65N15PII. S00361429024049231. Introduction. We are concerned with nonconforming finite element methods for second-order elliptic problems. Nonconforming elements have been used effectively especially in fluid and solid mechanics due to their stability. Recently, these elements have attracted increasing attention from scientists and engineers in more wide areas, as this type of element is potentially useful in parallel computing.The use of finite elements for Stokes problems, which is fundamental in fluid mechanics, usually requires the discrete Babuska-Brezzi condition (inf-sup condition) to be satisfied by the velocity and pressure variables, generally set in the mixed finite element formulation; for instance, the standard P 1 -P 0 pair for triangular decompositions or the Q 1 -P 0 pair for quadrilateral decompositions of the computational domain lead to checkerboard solutions for pressure. However, if the nonconforming elements introduced in [3,8,15,5] are used to approximate the velocity part instead of the usual P 1 or Q 1 elements, the Babuska-Brezzi condition is easily satisfied, and thus stable solutions are obtained. Nonconforming finite element methods have been proved to be effective for several parameter dependent elasticity problems in a stable fashion such that the methods converge independently of the Lamé parameters, while standard conforming methods fail to converge as the parameters tend to a locking limit; see [2,12,13].Moreover, in view of domain decomposition methods, the use of nonconforming elements facilitates the exchange of information across each subdomain and provides spectral radius estimates for the iterative domain decomposition operator [9].The nonconforming simplicial finite element space of lowest degree introduced by Crouzeix and Raviart [8] is identical to the corresponding conforming one (that

In magnetic resonance electrical impedance tomography (MREIT), we measure the induced magnetic flux density inside an object subject to an externally injected current. This magnetic flux density is contaminated with noise, which ultimately limits the quality of reconstructed conductivity and current density images. By analysing and experimentally verifying the amount of noise in images gathered from two MREIT systems, we found that a carefully designed MREIT study will be able to reduce noise levels below 0.25 and 0.05 nT at main magnetic field strengths of 3 and 11 T, respectively, at a voxel size of 3 x 3 x 3 mm(3). Further noise level reductions can be achieved by optimizing MREIT pulse sequences and using signal averaging. We suggest two different methods to estimate magnetic flux noise levels, and the results are compared to validate the experimental setup of an MREIT system.

Magnetic resonance current density imaging (MRCDI) provides a current density image by measuring the induced magnetic flux density within the subject with a magnetic resonance imaging (MRI) scanner. Magnetic resonance electrical impedance tomography (MREIT) has been focused on extracting some useful information of the current density and conductivity distribution in the subject Omega using measured B(z), one component of the magnetic flux density B. In this paper, we analyze the map Tau from current density vector field J to one component of magnetic flux density B(z) without any assumption on the conductivity. The map Tau provides an orthogonal decomposition J = J(P) + J(N) of the current J where J(N) belongs to the null space of the map Tau. We explicitly describe the projected current density J(P) from measured B(z). Based on the decomposition, we prove that B(z) data due to one injection current guarantee a unique determination of the isotropic conductivity under assumptions that the current is two-dimensional and the conductivity value on the surface is known. For a two-dimensional dominating current case, the projected current density J(P) provides a good approximation of the true current J without accumulating noise effects. Numerical simulations show that J(P) from measured B(z) is quite similar to the target J. Biological tissue phantom experiments compare J(P) with the reconstructed J via the reconstructed isotropic conductivity using the harmonic B(z) algorithm.

Magnetic resonance electrical impedance tomography (MREIT) aims at producing high-resolution cross-sectional conductivity images of an electrically conducting object such as the human body. Following numerous phantom imaging experiments, the most recent study demonstrated successful conductivity image reconstructions of postmortem canine brains using a 3 T MREIT system with 40 mA imaging currents. Here, we report the results of in vivo animal imaging experiments using 5 mA imaging currents. To investigate any change of electrical conductivity due to brain ischemia, canine brains having a regional ischemic model were scanned along with separate scans of canine brains having no disease model. Reconstructed multi-slice conductivity images of in vivo canine brains with a pixel size of 1.4 mm showed a clear contrast between white and gray matter and also between normal and ischemic regions. We found that the conductivity value of an ischemic region decreased by about 10-14%. In a postmortem brain, conductivity values of white and gray matter decreased by about 4-8% compared to those in a live brain. Accumulating more experience of in vivo animal imaging experiments, we plan to move to human experiments. One of the important goals of our future work is the reduction of the imaging current to a level that a human subject can tolerate. The ability to acquire high-resolution conductivity images will find numerous clinical applications not supported by other medical imaging modalities. Potential applications in biology, chemistry and material science are also expected.

SUMMARYThe objective of the present study is to show that the numerical instability characterized by checkerboard patterns can be completely controlled when non-conforming four-node ÿnite elements are employed. Since the convergence of the non-conforming ÿnite element is independent of the Lamà e parameters, the sti ness of the non-conforming element exhibits correct limiting behaviour, which is desirable in prohibiting the unwanted formation of checkerboards in topology optimization. We employ the homogenization method to show the checkerboard-free property of the non-conforming element in topology optimization problems and verify it with three typical optimization examples.