The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L^2, and satisfies the stable decomposition property.

翻译：泡沫变换是一种将微分形式（这些形式关于给定区域三角剖分分段光滑）分解为局部泡沫之和的过程。本文提出了作者先前提出的de Rham复形框架中构造的改进版本。该分解的主要改进在于：与先前结果中单个泡沫为有理函数且局部泡沫组之和保持分段光滑性不同，新分解在严格意义上具有保空间性质，即每个局部泡沫均保持分段光滑性。该变换的一个重要特性是其构造仅依赖于给定区域的三角剖分，与任何有限元空间无关。另一方面，所有标准的分段多项式空间在该变换下保持不变。该变换的其他关键性质包括：与外微分可交换、在L^2中具有有界性，并满足稳定分解性质。