Abstract
Abstract In this article, we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems, it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the blockwise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.
Highlights
Methods for the solution of boundary value problems of elliptic partial dierential equations typically involve non-local operators
In this article we propose to construct the hierarchical matrix approximation in a block-adaptive way, i.e., in addition to the adaptivity of adaptive cross approximation (ACA) on each block we add another level of adaptivity to the construction of the hierarchical matrix
If we extend the L2-scalar product to a duality pairing between H−1/2(∂Ω) and H1/2(∂Ω), the boundary integral equation (2) can be stated in variational form as (Vψ, ψ )L2 = (
Summary
Methods for the solution of boundary value problems of elliptic partial dierential equations typically involve non-local operators. In addition to reducing the numerical eort for assembling and storing the matrix, this approach has the practical advantage that the block-wise accuracy of ACA is not a required parameter any more Choosing this parameter in usual ACA approximations is not obvious as the relation between the block-wise accuracy and the error of the solution usually depends on the respective problem. Numerical examples will be presented at the end of the article which show that the new method is more ecient than ACA with respect to both computational time and required storage in situations where the solution in some sense contains structural dierences or situations that result from locally over-rened meshes
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