Uniform preconditioners for problems of positive order

Raymond van Veneti\"e; Rob Stevenson

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Uniform preconditioners for problems of positive order

Rob Stevenson, Raymond van Venetië

Numerical Analysis · Submitted 21 Jun 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Using the framework of operator or Cald\'{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s \in [0,2]$ discretized with continuous finite (or boundary) elements.

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Using the framework of operator or Cald\'{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s \in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Using the framework of operator or Cald\'{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s \in [0,2]$ discretized with continuous finite (or boundary) elements.	contribution 0.93	0.86	0.66	asserted 1.00
s1	The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity.	result 0.83	0.78	0.22	asserted 1.00
s2	Herewith the construction of a so-called dual mesh is avoided.	contribution 0.66	0.75	0.09	asserted 1.00

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Original abstract

Using the framework of operator or Cald\'{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s \in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper

We mention multigrid preconditioners, [ref] based on uniformly refined mesh hierarchies and operator preconditioning, [ref] , which requires one to realize an operator of the opposite order.

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion · Introduction · 0.94

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Builds on 1 paper

It will build on our experiences with this approach for problems of negative order reported in [ref] .

Uniform preconditioners for problems of negative order · Introduction · 0.99

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To circumvent (explicit) dependence on the refined partition $_*$, we shall apply the idea described in [ref] .

Uniform preconditioners for problems of negative order · Construction of $B_∈(_, _')$ · 0.98

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