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This work focuses on steady and unsteady Navier-Stokes equations in a reduced order modeling framework. The methodology proposed is based on a Proper Orthogonal Decomposition within a levelset geometry description and the problems of…
Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier…
We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
A central question in numerical homogenization of partial differential equations with multiscale coefficients is the accurate computation of effective quantities, such as the homogenized coefficients. Computing homogenized coefficients…
We analyze the approximation by mixed finite element methods of solutions of equations of the form $-\mbox{div\,} (a\nabla u) = g$, where the coefficient $a=a(x)$ can degenerate going to cero or infinity. First, we extend the classic error…
This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed…
This paper introduces the approach of Randomized Orthogonal Decomposition (ROD) for producing twin data models in order to overcome the drawbacks of existing reduced order modelling techniques. When compared to Fourier empirical…
A new approach to model order reduction of the Navier-Stokes equations at high Reynolds number is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier-Stokes…
A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous…
We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure…
This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding…
The dynamics of two-component solitons is studied, analytically and numerically, in the framework of a system of coupled extended nonlinear Schr\"odinger equations, which incorporate the cross-phase modulation,…
This paper concerns the numerical solution of three-dimensional degenerate Kawarada equations. These partial differential equations possess highly nonlinear source terms, and exhibit strong quenching singularities which pose severe…
We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to…
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study…
In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of…
Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…
We develop a new multiscale finite element method for Laplace equation with oscillating Neumann boundary conditions on rough boundaries. The key point is the introduction of a new boundary condition that incorporates both the…
This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis --- which has been made possible by recent contributions…
This paper introduces a reduced-order modeling approach based on finite volume methods for hyperbolic systems, combining Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and Proper Interval…