We introduce a family of mixed methods and discontinuous Galerkin discretisations designed to numerically solve the Oseen equations written in terms of velocity, vorticity, and Bernoulli pressure. The unique solvability of the continuous problem is addressed by invoking a global inf-sup property in an adequate abstract setting for non-symmetric systems. The proposed finite element schemes, which produce exactly divergence-free discrete velocities, are shown to be well-defined and optimal convergence rates are derived in suitable norms. This mixed finite element method is also pressure-robust. In addition, we establish optimal rates of convergence for a class of discontinuous Galerkin schemes, which employ stabilisation. A set of numerical examples serves to illustrate salient features of these methods.

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Mathai’s pathway model is playing an increasingly prominent role in statistical distributions. As a generalization of a great variety of distributions, the pathway model allows the studying of several non-linear dynamics of complex systems. Here, we construct a model, called the Pareto–Mathai distribution, using the fact that the earthquakes’ magnitudes of full catalogues are well-modeled by a Mathai distribution. The Pareto–Mathai distribution is used to study artificially induced microseisms in the mining industry. The fitting of a distribution for entire range of magnitudes allow us to calculate the completeness magnitude ($M_c$). Mathematical properties of the new distribution are studied. In addition, applying this model to data recorded at a Chilean mine, the magnitude $M_c$ is estimated for several mine sectors and also the entire mine.

We introduce a family of mixed methods and discontinuous Galerkin discretisations designed to numerically solve the Oseen equations written in terms of velocity, vorticity, and Bernoulli pressure. The unique solvability of the continuous problem is addressed by invoking a global inf-sup property in an adequate abstract setting for non-symmetric systems. The proposed finite element schemes, which produce exactly divergence-free discrete velocities, are shown to be well-defined and optimal convergence rates are derived in suitable norms. In addition, we establish optimal rates of convergence for a class of discontinuous Galerkin schemes, which employ stabilisation. A set of numerical examples serves to illustrate salient features of these methods.

We propose and analyse an augmented mixed finite element method for the Navier--Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition. The theoretical and practical implications of using augmentation is discussed in detail. In addition, we use fixed--point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.