Error estimates with the optimal convergence order are proved for a pressurestabilized characteristics finite element scheme for the Oseen equations. The scheme is a combination of Lagrange-Galerkin finite element method and Brezzi-Pitkäranta's stabilization method. The scheme maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The theoretical convergence order is recognized by two-and three-dimensional numerical results.Keywords Error estimates · The finite element method · The method of characteristics · Pressure-stabilization · The Oseen equations Mathematics Subject Classification 65M12 · 65M60 · 65M25 · 76D07

Optimal error estimates of stable and stabilized Lagrange-Galerkin (LG) schemes for natural convection problems are proved under a mild condition on time increment and mesh size. The schemes maintain the common advantages of the LG method, i.e., robustness for convection-dominated problems and symmetry of the coefficient matrix of the system of linear equations. We simply consider typical two sets of finite elements for the velocity, pressure and temperature, P2/P1/P2 and P1/P1/P1, which are employed by the stable and stabilized LG schemes, respectively. The stabilized LG scheme has an additional advantage, a small number of degrees of freedom especially for three-dimensional problems. The proof of the optimal error estimates is done by extending the arguments of the proofs of error estimates of stable and stabilized LG schemes for the Navier-Stokes equations in previous literature.

Nonlinear magnetization dynamics excited by spin-transfer effect with feedback current is studied both numerically and analytically. The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagnet. Transient behavior from chaotic to steady oscillation is also observed in another range of the feedback parameter. An analytical theory is also developed, which indicates the appearance of multiple attractors in a phase space due to the feedback current. An instantaneous imbalance between the spin-transfer torque and damping torque causes a transition between the attractors, and results in the complex dynamics.PACS numbers:

Fluids exist universally in nature and technology. Among the many types of fluid flows is the well-known vortex shedding, which takes place when a fluid flows past a bluff body. Diverse types of vortices can be found in this flow as the Reynolds number increases. In this study, we reveal that these vortices can be employed for conducting certain types of computation. The results from computational fluid dynamics simulations showed that optimal computational performance is achieved near the critical Reynolds number, where the flow exhibits a twin vortex before the onset of the Kármán vortex shedding associated with the Hopf bifurcation. It is revealed that as the Reynolds number increases toward the bifurcation point, the input sensitivity of the twin vortex motion also increases, suggesting the modality of information processing within the system. Our finding paves a novel path to understand the relationship between fluid dynamics and its computational capability.