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Uniform preconditioners for problems of negative order
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Uniform preconditioners for operators of negative order discretized by (dis)continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears. Besides the cost of the application of the latter discretized operator, the other cost of the preconditioner scales linearly with the number of mesh cells. Compared to earlier proposals, the preconditioner has the following advantages: It does not require the inverse of a non-diagonal matrix; it applies without any mildly grading assumption on the mesh; and it does not require a barycentric refinement of the mesh underlying the trial space.
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An Auxiliary Space Preconditioner for Fractional Laplacian of Negative Order
Auxiliary space preconditioners of the form ∇*B_div∇ are shown to be uniformly spectrally equivalent to the negative-order fractional Laplacian; the required fractional H(div) preconditioner is built by additive multi...
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