Talk:Finite element method

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    Reviewer A

    The article provides a concise introduction to the Finite Element (FE) Method explaining the two main concepts: construction of basis functions and the weak formulations. The text, written by a forefather of the FE Method, is hard to improve and I definitely approve it for publication.

    Suggested changes:

    I would use directly the concept of barycentric (affine, area) coordinates when introducing the FE shape functions rather that the explicit formulas for them.

    The author has swept under the carpet the concept if DISCRETE inf-sup constant and discrete stability. According to the Babuska' Theorem, only the discrete stability and approximability imply the convergence. While for a class of coercive problems including the discussed example, the stability on the continuous level implies the discrete one, in general this is not true. In fact, the main business of the numerical analysis for the last 50 years, has been to construct schemes that satisfy the discrete inf-sup condition (mixed approximations and Brezzi theory, exact sequences etc.) I would at least mention the concept of the discrete inf-sup condition.

    Over the years, a number of other energy spaces than classical Sobolev spaces, has been introduced, e.g. spaces H(div), H(curl) and the corresponding conforming elements of Raviart-Thomas and Nedelec. I would perhaps emphasize that the Sobolev space is a particular (historical and most important) choice that corresponds to the discussed example only.

    Minor changes introduced in the text:

    aligned equality signs in formula (1) and added a curly brace

    the outward normal unit vector

    notation for partial derivatives changed

    have increased the size of integrals in (6)

    have changed ``some to ``all after eq.(7)

    notation in (8) corrected

    (1) The opening statement "The Finite Element Method (FEM) is arguably the most powerful method known for the numerical solution of boundary- and initial- value problems characterized by partial differential equations." is a bit of an overstatement. I would argue that the FEM is one of several well established methods such as finite difference, finite volume, spectral, weighted residual, Galerkin, least squares and meshless methods, and I suggest that an opening statement that places the FEM in the overall context of methods for IV and BV problems would be better.

    (2) Scholarpedia articles are, I think, intended to be introductory. The FEM article has rather substantial abstract mathematics, and I think at least one basic, introductory example application would be helpful, giving the details of the FEM discretization and the errors resulting from it, with possibly a link to a code, e.g., in Fortran or Matlab, that the reader could execute to gain some direct experience with the FEM.

    In response to the author:

    (1) Perhaps the author could indicate the many variants of the FEM to provide the reader with some additional background.

    (2) Generally an encyclopedia is a starting point for the reader and therefore an introductory aspect of the article is appropriate. For example, an introductory application would be a useful addition for getting the reader started in FEM analysis (and a page limit is not a consideration or restriction).

    Reviewer B

    The paper is well written. I have the following minor comments

    1. Page 1. Instead \(W^{p,m}(\Omega )\) it would be better to use standard notation \(W^{m,p}\).

    2. Page 1. Before formula (1) underline that the function \(u\) is a scalar function.

    3. Page 2. In the formula (3), write \(\| v\|_U\) instead \(\| v\|_V\).

    4. Page 3 line 10 top. The last term is \(\|u\|_{W^{t,s}(\Omega )}\), it should be \(\| \bullet \|_{W^{t,s}(\Omega_k)}\).

    5. Page 4 Line 15 top. Better to use directly \(t\) and delete \(q = t\).

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