Numerical Models for Differential Problems

By Alfio Maria Quarteroni. 2009

604 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/651ec7eb1be4876d630bfdad64c1c08a4ff421da

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In Applied Mathematics

In Numerical Analysis

Citations in the corpus

Shape optimization for viscous flows by reduced basis methods and free‐form deformation (2012, neutral): “Geometry description, state equations and system observation To model the incoming branch of a bypass we consider a longitudinal section in the mean plane and the corresponding two-dimensional domain Ωo(μ); the steady Stokes equations [32] in Ωo(μ) read:  −ν∆v(μ) +∇p(μ) = f in Ωo(μ) ∇ · ”
Geodesic Convolutional Shape Optimization (2018, methodology): “Traditional ones rely on numerical discretization of the Navier-Stokes Equations (NSE) using ﬁnite-difference, ﬁnite-element, or ﬁnite volume methods (Quar-teroni & Quarteroni, 2009; Skinner & Zare-Behtash, 2018).”
Geometric Deep Learning for Volumetric Computational Fluid Dynamics (2019, methodology): “For more informations about these methods, see [13].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “Equation (31) – Backward Euler-Galerkin discretization of (8) – shall be our point of departure: we shall presume that ∆t is sufficiently small and N is sufficiently large such that uN (tk;µ) and sN (tk;µ) are effectively indistinguishable from u(tk; µ) and s(tk;µ), respectively.”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, neutral): “Generally Navier-Stokes equations are written as Fx(u(x); ν, p) = −div[ν(∇u+∇uT )] + (u+∇)u+∇p− f , where, u is the speed of the fluid, p the pressure and ν the viscosity (Quarteroni, 2013).”
The Schwarz alternating method in solid mechanics (2017, methodology): “Elimination of the Schwarz Boundaries Degrees of Freedom In the domain decomposition community it is common to altogether eliminate the degrees of freedom on the Schwarz boundaries Γi by explicitly introducing the projection (27) into the linearization of the residual (21) [57, 64, 40, 50].”
Sparse-grid, reduced-basis Bayesian inversion (2015, methodology): “For this purpose, we rely on a reduced basis method [42, 46] in approximating a high-fidelity PG solution.”
Isogeometric Analysis and error estimates for high order partial differential equations in Fluid Dynamics (2014, methodology): “14) where the bilinear form a∗ : V × V → R is defined as a∗(v, w) := a(w, v) for all w, v ∈ V (see for instance [29, 35, 40]).”