By Olli Mali
The value of accuracy verification equipment was once understood on the very starting of the improvement of numerical research. contemporary a long time have obvious a speedy development of effects with regards to adaptive numerical equipment and a posteriori estimates. in spite of the fact that, during this vital sector there usually exists a seen hole among mathematicians growing the speculation and researchers constructing utilized algorithms that may be utilized in engineering and clinical computations for assured and effective errors control.
The pursuits of the publication are to (1) supply a clear rationalization of the underlying mathematical concept in a method available not just to complicated numerical analysts but in addition to engineers and scholars; (2) current exact step by step algorithms that persist with from a concept; (3) speak about their merits and disadvantages, parts of applicability, supply strategies and examples.
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Extra info for Accuracy Verification Methods: Theory and Algorithms
Our goal is to obtain a guaranteed (fully reliable) estimate of u − v in terms of the energy norm |u − v |2 := ξ2 2 a u −v + b(u − v)2 dx. 4) ξ1 for any trial function w ∈ V0 , where V0 contains the functions from H 1 (Ω) vanishing at ξ1 and ξ2 (see Sect. 1). 4) in the equivalent form ξ2 ξ2 a(u − v) w + b(u − v)w dx = ξ1 f w − av w − bvw dx. 5) ξ1 Let y(x) be an arbitrary function in H 1 (Ω). 5) as follows: ξ2 f w − av w − bvw dx = ξ1 ξ2 f + y − bv w − av − y w dx. ξ1 Now, we set w = u − v and obtain |u − v |2 = ξ2 f + y − bv (u − v) − av − y (u − v) dx.
10, we schematically depict the space V , the subspaces Vh , Wh , and Vhref and the corresponding approximate solutions uh and uhref . It is easy to see that 2 2 ∇(u − uh ) dx = Ω Ω ∇(u − uhref ) dx + +2 Ω 2 Ω ∇(uh − uhref ) dx ∇(u − uhref ) · ∇(uhref − uh ) dx, where Ω ∇(u − uhref ) · ∇(uhref − uh ) dx = Ω f (uhref − uh ) dx − Ω f uhref dx + f uh dx = 0. Ω Hence, ∇(u − uh ) 2 = ∇(u − uhref ) 2 + ∇(uh − uhref ) = ∇(u − uhref ) 2 I Runge (uh ) . 50) which formalizes a rather natural condition: uhref is closer to u than uh .
1 Averaging of Gradients (Fluxes) Gradient averaging procedures are often used to post-process gradients (fluxes, stresses) computed by finite element approximations of elliptic boundary value problems. Among first publications in this direction we mention the papers [ZZ87, ZZ88], which generated an interest in gradient recovery methods. , [AO92, BC02, BR93, BS01, HTW02, Ver96, Wan00, WY02, ZBZ98, ZN05]). , [KN84, KNS98, Wah95]). Superconvergence arises on regular (quasiregular) meshes and, in simple terms, means that some components of approximate solutions obtained by inexpensive post-processing procedures converge to the corresponding components of the exact solution with a rate higher than the rate that can be predicted by standard a priori estimates.
Accuracy Verification Methods: Theory and Algorithms by Olli Mali