Analysis Of The Finite Element Method

By S. Blau. 2016

818 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a4e5ba40c438056af000e5d18894c694cfd15f69

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In Civil and Structural Engineering

In Computational Theory and Mathematics

Citations in the corpus

On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes (2013, methodology): “1 of [28]), provided that ah is uniformly coercive on V 0 h , we have the following bound for this term:”
Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis (2016, methodology): “Nevertheless, as is commonly done in finite element theory, we use Gauss rules that are optimal for the smallest higher odddegree polynomials [11, 16, 44].”
Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis (2017, background): “The basic idea, explained in the following example, is very similar to that of the patch test(3) [25, 47] and applies only to the leading terms in a⋆(·, ·), involving first derivatives (case where α = 0).”
Bayesian Probabilistic Numerical Methods (2017, methodology): “For Gaussian belief distributions μ, a detailed theoretical analysis of existence of ACA optimal methods was provided in Tarieladze and Vakhania [2007]. Here we prove that optimal information for a Bayesian PNM is, under a constraint on the loss function L, the same as optimal information for ACA.”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, methodology): “Optimization-based methods have been popular for some time in the form of leastsquares finite element methods (LSFEM) [1, 2, 3], and element-free Galerkin methods [4, 5], where even for mesh-free formulations, theoretical convergence criteria have been examined [6].”
The nonconforming virtual element method (2014, background): “Nonconforming finite elements were first recognized as a variational crime; a term first coined by Strang in [40, 41].”
Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence (2018, methodology): “In view of (27), this is consistent with the known fact that the biharmonic equation must be solved with piecewise polynomials of at least cubic degree to obtain an optimal rate of convergence in L 2 ; see, e.g., p. 118 in [31].”
Reduced integration at superconvergent points in isogeometric analysis (2018, background): “Therefore reduced and optimal quadrature schemes already well established for FEMs, see Strang & Fix [2], have attracted revitalized attention in IGA, e.”
Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis (2014, methodology): “Following [27, 38], the complete degree of quadrature accuracy required to satisfy criterion (7) for full tensor-product basis functions can be determined by the relation (d+1)p− 2m.”
Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth (2017, background): “Whilst enrichment itself is an idea dating back to the 1970’s [16], and is also found in the meshfree literature since the early 1990’s [17]2 partition of unity enrichment enables a local enrichment of the basis, leading to banded matrices and enabling to flexibly combine different enrichments withi”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, neutral): “If the essential boundary conditions are exactly met, then this precludes ‘variational crimes’ in a Ritz method [68].”
Integration by interpolation and look-up for Galerkin-based isogeometric analysis (2015, background): “More precisely, it is sufficient to verify that the bilinear form in (14) satisfies the inequality ã(vh, vh) ≥ µ|vh|21,Ω ∀ vh ∈ Vh , where | · |1,Ω is the H1−seminorm on Ω (which is here a norm due to the homogeneous Dirichlet boundary) and the constant µ is independent of the mesh size h.”
Isogeometric Analysis and error estimates for high order partial differential equations in Fluid Dynamics (2014, methodology): “14) satisfies the hypotheses of the Nec̆as–Babus̆ka Theorem [35, 41, 43]; thus, there exists a unique solution φF ∈ V of the adjoint problem (3.”
The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems (2017, background): “The solution to this problem suffers from the well-known “corner singularity” caused by the nature of the domain [11].”
The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas (2020, background): “[146], where h is the element size, and p is the polynomial order of the finite elements.”
Numerical Methods for Elliptic and Parabolic Partial Differential Equations (2003, neutral): “[31] G.”