PHYSICS-BASED BALANCING DOMAIN DECOMPOSITION BY CONSTRAINTS FOR HETEROGENEOUS PROBLEMS
Résumé
In this work, we present a balancing domain decomposition by constraints method based on an aggregation of elements depending on the physical coefficients. Instead of imposing constraints on purely geometrical objects (faces, edges, and vertices) of the partition interface, we use interface objects (subfaces, subedges, and vertices) determined by the variation of the coefficients. The new method is easy to implement and does not require to solve any eigenvalue or auxiliary problem. When the physical coefficient in each object is constant at every subdomain containing the object, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient. The constant coefficient condition is possible for multi-material problems. However, for heterogeneous problems with coefficient varying across a wide spectrum of values in a small spatial scale, such restriction might result in too many objects (a large coarse problem). In this case, we propose a relaxed version of the method where we only require that the maximal contrast of the physical coefficient in each object is smaller than a predefined threshold. The threshold can be chosen so that the condition number is reasonably small while the size of the coarse problem is not too large. An extensive set of numerical experiments is provided to support our findings.
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