# An adaptive BDDC algorithm in variational form for mortar discretizations

## TL;DR

This paper introduces an adaptive BDDC algorithm in variational form for high-order mortar discretizations of 2D elliptic problems, demonstrating robustness and efficiency across various models.

## Contribution

It presents a simplified adaptive BDDC algorithm for mortar methods that avoids continuity constraints at vertices, with proven condition number bounds independent of mesh size and coefficient contrast.

## Key findings

- Condition number bounded by user tolerance and interface count
- Algorithm is robust for high-order mortar discretizations
- Numerical tests confirm efficiency and robustness

## Abstract

A balancing domain decomposition by constraints (BDDC) algorithm with adaptive primal constraints in variational form is introduced and analyzed for high-order mortar discretization of two-dimensional elliptic problems with high varying and random coefficients. Some vector-valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm, and the coarse space is formed by using a transformation operator on each interface. Compared with the adaptive BDDC algorithms for conforming Galerkin approximations, our algorithm is more simple, because there is not any continuity constraints at subdomain vertices in the mortar method involved in this paper. The condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain, and independent of the mesh size and the contrast of the given coefficients. Numerical results show the robustness and efficiency of the algorithm for various model problems.