Domain Decomposition Methods (DDM) based on Schwarz methods are widely used as efficient preconditioners for large linear systems in parallel computing. Their robustness typically relies on the introduction of a coarse space, enabling scalability with respect to both problem parameters and the number of subdomains. While these techniques are well established for partial differential equations, significantly less work has been devoted to non-local problems such as boundary integral equations (BIEs) or the fractional Laplacian.
Non-local problems pose additional challenges, both theoretically and computationally: the global nature of the underlying kernels complicates the derivation of rigorous bounds on the spectral properties of the preconditioned system, while their discretization leads to dense matrices requiring efficient compression techniques.
In this talk, we present recent results on two-level Schwarz preconditioners for non-local problems, together with the lightweight library Htool-DDM, designed to be integrated into larger simulation frameworks to provide compression techniques and domain decomposition solvers for non-local operators. Applications to BIEs using FreeFEM and to the fractional Laplacian using PyNucleus will be demonstrated, illustrating the flexibility and practical performance of the approach.

Many software systems for FEM utilize a global interface that matches FEM and functional analysis textbooks. However, when we lower global FEM concepts to parallel loops and arrays, we do so in a patchwork fashion, dispatching on functional analytic concepts such as the continuity of function spaces and the class of integrals. We end up with duplicated code/codegen as well as gaps in functionality. In our work building FEM software parameterized on element definitions, we frequently encountered these issues. Thus, we propose a new set of intermediate abstractions, realized as an intermediate representation, that model the discrete consequences - the code generation consequences - of our numerical mathematics. Our IR models integrated pointwise tensor expressions on meshes (IPTEM). We generate code based on the relationship between pointwise computations, the piecewise continuity of generalized functions, and mesh topologies as relations. We design a code generation algorithm, topological enumeration of integrated pointwise tensor expressions on meshes, that handles all code generation in our system by analyzing the aforementioned relationship. We design a second abstraction, realized as a global data structure, the single valued assembled operator, which represents a well defined realization of IPTEMs as a structured sparse tensor. By combining our two abstractions, we can design interpolation and elimination boundary conditions operations that are more expressive and capable. This work is in progress though as we do not yet have key ideas, such as MPI support, fusion, and additional operations on SVAO.

We are concerned with the efficient numerical approximation of solutions of the time-domain linear Maxwell system. Methods like the classical finite-difference time-domain method or finite integration techniques rely on the approximation of the electric and the magnetic field on two interlaced (Cartesian) grids respectively.
Our method expands this idea to general tetrahedral meshes by defining a dual grid using the barycentric subdivision of the primal mesh, resulting in a decomposition into hexahedra.
Contrary to previous approaches we use Lagrangian polynomial basis functions with respect to tensor product integration points on the unit cube that are subsequently re-mapped to the hexahedra which form the physical elements of the primal and dual grids respectively. This approach provides generically computable, stable high-order bases.
We use leap frog time-stepping (or high-order variants thereof) for the time discretization which only requires the application of the discrete differential operators and the inverse mass matrices in each time step. Using lumped mass matrices with respect to the defining points of the basis functions leads to block diagonal mass matrices with block-size independent of the polynomial degree.
Numerical experiments underline the facts that the resulting algorithm provides converging, spurious-free solutions and is efficient, compared to competing methods.

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