An efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. It is proved that at each time step the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within a finite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm.

Purpose: Design, implement, integrate, and characterize a customized coil system that allows for generating spatial encoding magnetic fields (SEMs) in a highly‐flexible fashion.Methods: A gradient coil with a high number of individual elements was designed. Dimensions of the coil were chosen to mimic a whole‐body gradient system, scaled down to a head insert. Mechanical shape and wire layout of each element were optimized to increase the local gradient strength while minimizing eddy current effects and simultaneously considering manufacturing constraints.Results: Resulting wire layout and mechanical design is presented. A prototype matrix gradient coil with 12 × 7 = 84 elements consisting of two element types was realized and characterized. Measured eddy currents are <1% of the original field. The coil is shown to be capable of creating nonlinear, and linear SEMs. In a DSV of 0.22 m gradient strengths between 24 mT∕m and 78 mT∕m could be realized locally with maximum currents of 150 A. Initial proof‐of‐concept imaging experiments using linear and nonlinear encoding fields are demonstrated.Conclusion: A shielded matrix gradient coil setup capable of generating encoding fields in a highly‐flexible manner was designed and implemented. The presented setup is expected to serve as a basis for validating novel imaging techniques that rely on nonlinear spatial encoding fields. Magn Reson Med 79:1181–1191, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

An element-local $L^2$-projected $C^0$ finite element method is presented to approximate the nonsmooth solution being not in $H^1$ of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local $L^2$ projectors are applied to both curl and div operators. The $C^0$ linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in $L^2$ norm is established uniform in the mesh-size, and the condition number ${\cal O}(h^{-2})$ of the resulting linear system is proven. For the solution and its curl in $H^r$ with $r<1$ we obtain an error bound ${\cal O}(h^r)$ in an energy norm. Numerical experiments confirm the theoretical error bound.

Purpose: A multi‐coil shim setup is designed and optimized for human brain shimming. Here, the size and position of a set of square coils are optimized to improve the shim performance without increasing the number of local coils. Utilizing such a setup is especially beneficial at ultrahigh fields where B₀ inhomogeneity in the human brain is more severe.Methods: The optimization started with a symmetric arrangement of 32 independent coils. Three parameters per coil were optimized in parallel, including angular and axial positions on a cylinder surface and size of the coil, which were constrained by cylinder size, construction consideration, and amplifiers specifications. B₀ maps were acquired at 9.4T in 8 healthy volunteers for use as training data. The global and dynamic shimming performance of the optimized multi‐coil were compared in simulations and measurements to a symmetric design and to the scanner's second‐order shim setup, respectively.Results: The optimized multi‐coil performs better by 14.7% based on standard deviation (SD) improvement with constrained global shimming in comparison to the symmetric positioning of the coils. Global shimming performance was comparable with a symmetric 65‐channel multi‐coil and full fifth‐order spherical harmonic shim coils. On average, an SD of 48.4 and 31.9 Hz was achieved for in vivo measurements after global and dynamic slice‐wise shimming, respectively.Conclusions: An optimized multi‐coil shim setup was designed and constructed for human whole‐brain shimming. Similar performance of the multi‐coils with many channels can be achieved with a fewer number of channels when the coils are optimally arranged around the target.