Nitsche's method for the stationary Boussinesq system under mixed and nonlinear boundary conditions

TL;DR

This paper analyzes Nitsche's finite element method for the stationary Boussinesq system with complex boundaries, establishing robustness, convergence, and error estimation, validated by numerical tests.

Contribution

The paper provides a rigorous analysis of Nitsche's method for complex boundary conditions in the Boussinesq system, including stability, convergence, and error estimation.

Findings

Robustness of the finite element scheme in complex boundaries.

Optimal convergence rates for approximation errors.

Reliability of residual-based a posteriori error estimators.

Abstract

In this paper we analyze Nitsche's method for the stationary Boussinesq system with Navier's slip and a nonlinear boundary condition. Our analysis of the formulation establishes the robustness of a finite elements scheme in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using fixed-point theorems under a standard smallness assumption on the data. We also provide optimal convergence rates for the approximation error. Furthermore, the efficiency and reliability of residual-based a posteriori error estimators are established. We validate our theory through several numerical tests.