PACE solver description: tdULL

We describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. An implementation was submitted to the exact track of PACE 2020. tdULL is a branch and bound algorithm branching on inclusion-minimal separators

Uniform preconditioners of linear complexity for problems of negative order

We propose a multi-level type operator that can be used in the framework of operator (or Caldéron) preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions. The cost of applying this multi-level operator scales linearly in the number of mesh cells. Therefore, it provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Uniform preconditioners for problems of negative order, Math. Comp. 89 (2020), 645–674]

Adaptive space-time BEM for the heat equation

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted $L_2$-norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$. While the results do not depend on the local aspect ratio between time and space, assuming the scaling $|J| \eqsim {\rm diam}(K)^2$ for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional $L_2$-terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate