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Using Finite Element Analysis to obtain a Weak form of a PDE, what is the difference between the two cases:

  1. A single element with a quadratic shape function
  2. Two elements with linear shape functions.

Thank you for any insights you can provide, Cameron

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  • $\begingroup$ This question is a bit vague because the answer will depend on the dimensionality of the problem and the exact form of the interpolation functions being used. But, roughly, a quadratic/linear function will be able to exactly express solutions that are quadratic/linear. This idea is called "completeness" in the FE literature (see www4.hcmut.edu.vn/~tttruong/IFEM.Ch19.Slides.pdf). You can approximate a quadratic function by a set of linear functions, but whether two will suffice depends on the particular quadratic in question. $\endgroup$ Oct 24, 2021 at 19:52
  • $\begingroup$ Element and shape function choice in FEM depend on the problem's geometry. A quadratic shape gives accuracy for curved boundaries, while two linear elements are best for high aspect ratios. $\endgroup$ Mar 19, 2023 at 1:02

1 Answer 1

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The shape of a quadratic function is a parabola. Parabolas have the equation f(x) = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The value of a determines the width and the direction of the parabola, while the vertex depends on the values of a, b, and c.

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A linear function is a function which forms a straight line in a graph. It is generally a polynomial function whose degree is utmost 1 or 0. Examples:

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