# Computing with Hp-Adaptive Finite Elements, Vol. 2: by Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej

By Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszynski, Waldemar Rachowicz, Adam Zdunek

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Additional info for Computing with Hp-Adaptive Finite Elements, Vol. 2: Frontiers: Three Dimensional Elliptic and Maxwell Problems with Applications

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25) The 1/r weight in front of the ur component is a consequence of the formula for the θ θ . For finite element basis functions, the assumption on finite energy leads to the following condition: ur = 0 at r = 0 and it is frequently misinterpreted as a boundary condition on ∂ . The r = 0 part of the boundary of the “trace domain” does not correspond to any part of the boundary of the original domain, and any condition stated there is a result of the finite energy assumption only. 3 Maxwell Equations For an introduction to Maxwell equations, we refer to the first volume of this book.

43) ∂ −→ P p−1 −→ {0} Here ∂ stands for the derivative operator. 44) Determining the interpolant involves solving a system of p − 1 linear equations. 44) is equivalent to the variational problem u2 ∈ P p , u2 (0) = u2 (1) = 0 (u2 , v) L 2 ( I ) = (u − u1 , v) L 2 ( I ) , ∀v ∈ P p : v(0) = v(1) = 0 Representing both u2 and test function v in terms of p − 1 bubble shape functions χ j , j = 1, . . , p − 1 p−1 p−1 j u2 χ j (x), u2 (x) = v(x) = j=1 vi χi (x) i=1 we obtain an equivalent system of p − 1 linear equations  j  Find u2 , j = 1, .

Finally, we comment on the choice of spaces for the axisymmetric problem. The finite energy assumption leads to a weighted Sobolev space with weights implied by the cylindrical coordinates. 25) The 1/r weight in front of the ur component is a consequence of the formula for the θ θ . For finite element basis functions, the assumption on finite energy leads to the following condition: ur = 0 at r = 0 and it is frequently misinterpreted as a boundary condition on ∂ . The r = 0 part of the boundary of the “trace domain” does not correspond to any part of the boundary of the original domain, and any condition stated there is a result of the finite energy assumption only.