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In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.

In this paper we propose an idea of constructing a macro--scale matrix system given a micro--scale matrix linear system. Then the macro--scale system is solved at cheaper computing costs. The method uses the idea of the generalized multiscale finite element method based. Some numerical results are presented.

In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We illustrate these properties in one, two and three spatial dimension, for quadrilateral elements and triangular elements, and for up to fourth order polynomials basis functions. To extend the method to non-linear problems, we introduce a linear approximation and show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.

In this article, a numerical scheme to find approximate solutions to the McKendrick-Von Foerster equation with diffusion (M-V-D) is presented. The main difficulty in employing the standard analysis to study the properties of this scheme is due to presence of nonlinear and nonlocal term in the Robin boundary condition in the M-V-D. To overcome this, we use the abstract theory of discretizations based on the notion of stability threshold to analyze the scheme. Stability, and convergence of the proposed numerical scheme are established.

In this paper, we construct a robust adaptive central-upwind scheme on unstructured triangular grids for two-dimensional shallow water equations with variable density. The method is well-balanced, positivity-preserving, and oscillation-free at the curve where two types of fluid merge. The proposed approach is an extension of the adaptive well-balanced, positivity-preserving scheme developed in Epshteyn and Nguyen (arXiv preprint arXiv:2011.06143, 2020). In particular, to preserve "lake-at-rest" steady states, we utilize the Riemann Solver with appropriately rotated coordinates to obtain the point values in the neighborhood of the fluid interface. In addition, to improve the efficiency of an adaptive method in the multifluid flow, the curve of density discontinuity is reconstructed by using the level set method and volume fraction method. To demonstrate the accuracy, high resolution, and efficiency of the new adaptive central-upwind scheme, several challenging tests for Shallow water models with variable density are performed.

We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary Method for spatial discretization is combined with an explicit predictor/multi-corrector time integration to integrate in time the numerical solutions to the shallow water equations, both for the full and reduced-order model. In order to improve the approximation of the solution manifold also for geometries that are untested during the offline stage, the snapshots have been pre-processed by means of an interpolation procedure that precedes the reduced basis computation. The methodology is tested on geometrically parametrized shapes with varying size and position.

The paper provides a novel framework to study the accuracy and stability of numerical integration schemes when employed for the time domain simulation of power systems. A matrix pencil-based approach is adopted to evaluate the error between the dynamic modes of the power system and the modes of the approximated discrete-time system arising from the application of the numerical method. The proposed approach can provide meaningful insights on how different methods compare to each other when applied to a power system, while being general enough to be systematically utilized for, in principle, any numerical method. The framework is illustrated for a handful of well-known explicit and implicit methods, while simulation results are presented based on the WSCC 9-bus system, as well as on a 1, 479-bus dynamic model of the All-Island Irish Transmission System.

We present and investigate a new type of implicit fractional linear multistep method of order two for fractional initial value problems. The method is obtained from the second order super convergence of the Gr\"unwald-Letnikov approximation of the fractional derivative at a non-integer shift point. The proposed method is of order two consistency and coincides with the backward difference method of order two for classical initial value problems when the order of the derivative is one. The weight coefficients of the proposed method are obtained from the Gr\"unwald weights and hence computationally efficient compared with that of the fractional backward difference formula of order two. The stability properties are analyzed and shown that the stability region of the method is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method. Numerical result and illustrations are presented to justify the analytical theories.

We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any poly-size $n$-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least $\log n/(2\log\log n)$ as $n$ grows. This is stronger than the known state of the art bounds of the form $\Omega(\log n/(k(n)\log\log n))$ for similar combinatorial optimization problems, where $k(n)$ depends on the optimality value. For example, for the largest clique problem $k(n)$ corresponds to the square of the size of the clique [Rossman 2010]. At the same time our results are not quite comparable since in our case the circuits are required to produce a solution itself rather than solving the associated decision problem. As in our earlier work, the approach is based on the overlap gap property (OGP) exhibited by random $p$-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.

In this paper, a numerical scheme for a nonlinear McKendrick-von Foerster equation with diffusion in age (MV-D) with the Dirichlet boundary condition is proposed. The main idea to derive the scheme is to use the discretization based on the method of characteristics to the convection part, and the finite difference method to the rest of the terms. The nonlocal terms are dealt with the quadrature methods. As a result, an implicit scheme is obtained for the boundary value problem under consideration. The consistency and the convergence of the proposed numerical scheme is established. Moreover, numerical simulations are presented to validate the theoretical results.