Approximate computation of expectations by Charles Stein

By Charles Stein

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Hanging node 1. 2. Fig. 10. Closures for avoiding hanging nodes in quadrilateral patch refinement. 3 Refinement Strategies in 2D 35 hanging node 1. 2. Fig. 11. 16 For uneven numbers of sub-cells in each patch cell, a convenient closure method can be derived that avoids triangular closures. 15 that can easily be generalized to higher uneven numbers. It should be mentioned, however, that for high numbers ν = 2k + 1, k ∈ N very thin quadrilaterals can occur which might degrade numerical stability. See fig.

For purely advection modeling, Kessler showed that a gradient based criterion results in even better approximation quality than a truncation error estimate [240, 393]. In general, proxy criteria often interleave with physics based criteria. A gradient or a curvature of a constituent is only a proxy, since not the gradient of the constituent causes the real problem but the low approximation quality of a numerical scheme. Karni and coworkers use a smoothness indicator for adaptive refinement control in a solver for hyperbolic systems [230].

1. e. vn = (v1 + v2 )/2. 2. Define new cells τ1 and τ2 by: τ1 = {v1 , vn , v3 }, τ2 = {vn , v2 , v3 }. 3. e. eτm1 = {v1 , v3 }, and eτm2 = {v2 , v3 }. This algorithm is depicted in fig. 2. Note that the order of vertices is important to maintain the orientation of triangles. The above definition maintains orientation in the daughter elements. 1, called bisection of longest edge. It differs from the above given algorithm in that it refines the longest edge only. When starting with a sufficiently regular initial triangle, both algorithms are equivalent.

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