Numerical Approximation of Partial Differential Equations

By Alfio Maria Quarteroni, A. Valli. 2008

2,298 citations recorded by the source corpus.

This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1d3ac10798abf585ef9a82bf3a779b246b5b37c5

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Citations in the corpus

An hp certified reduced basis method for parametrized parabolic partial differential equations (2011, positive): “Our problem can then be expressed in the form (4) with b(w, z, v;μ) = b1(w, v;μ)+ b2(w, z, v;μ) (we have used a skew-symmetric form of the non-linear convection operator b2(w, z, v;μ) to generate certain discrete stability properties [18]); note that all forms satisfy the ‘affine’ assumption.”
Reduced basis methods for Stokes equations in domains with non-affine parameter dependence (2009, background): “This problem has an inf–sup condition (LBB) [ 27 ]t o be”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, neutral): “Note that since `(·;μ) ∈ (L(Ω)) and u(·;μ) ∈ C((0, tf ); (L(Ω))) [51], s(t;μ) is indeed well-defined for all t ∈ I.”
Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation (2009, neutral): “, [109]): E Darcy ≤ CDh‖φ‖l CD > 0, with l = min(2, s− 1) if φ ∈ H(Ωp) (s ≥ 2), and E Stokes ≤ CSh(‖uf‖r+1 + ‖pf‖r) CS > 0, with r = 1, 2, provided the solution (uf , pf ) is regular enough so that the norms at the right hand side make sense.”
CESAM: a free code for stellar evolution calculations (2008, methodology): “A finite-elements method (see e.g. Quarteroni & Valli, 1994) is used to solve the diffusion equation.”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, methodology): “We note, however, that the RB methods we describe do not replace, but rather build upon and are measured (as regards accuracy) relative to, a finite element model [22,41, 126 ,147,158]: the reduced basis approximates not the exact solution but rather a “given” finite element discretization of (typic”
MAST-2D diffusive model for flood prediction on domains with triangular Delaunay unstructured meshes (2011, background): “This leads the velocity field satisfies the mass balance equation only in the weak sense [44].”
Finite difference/spectral approximations for the time-fractional diffusion equation (2007, neutral): “[3,17]): 8g 2 H ðKÞ, m P 1, 8gN 2 PN ðKÞ, ðg; rN Þ ðgN ; rN ÞN 6 ð2kg IN gk0 þ kg IN 1gk0ÞkrNk0 þ kg gNk0;NkrNk0;N 6 ðcN kgkm þ kg gNk0;N ÞkrNk0;N : Applying this result to g 1⁄4 uj; gN 1⁄4 uN for all j 1⁄4 1; 2; .”
Stream function formulation of surface Stokes equations (2018, methodology): “This formulation has been used in numerical simulations, although it has certain disadvantages related to boundary conditions and regularity ([17, 32]).”
A uniformly well-conditioned, unfitted Nitsche method for interface problems (2013, background): “The statement involves the union of triangles Gh 1,⌧ , Gh 2,⌧ defined in (7).”
Isogeometric Analysis for Topology Optimization with a Phase Field Model (2012, background): “[32, 54, 83]), requires the approximation of the design domain and disconnects the analysis, and hence, the optimization from its geometrical representation.”