High-order primal mixed finite element method for boundary-value correction on curved domain

TL;DR

This paper develops a high-order boundary correction method using Raviart-Thomas elements for solving boundary-value problems with Neumann conditions on curved domains, achieving optimal convergence rates.

Contribution

It introduces a high-order boundary value correction technique on curved domains using Raviart-Thomas elements, with proven convergence rates and numerical validation.

Findings

Achieves $O(h^{k+1/2})$ convergence in $L^2$ norm for velocity.

Achieves $O(h^k)$ convergence in $H^1$ norm for pressure.

Numerical experiments confirm theoretical convergence rates.

Abstract

This paper addresses the non-homogeneous Neumann boundary condition on domains with curved boundaries. We consider the Raviart-Thomas element (RTk ) of degree $k \geq 1 $on triangular mesh. on a triangular mesh. A key feature of our boundary value correction method is the shift from the true boundary to a surrogate boundary. We present a high-order version of the method, achieving an $O(h^k+1/2)$ convergence in $L^2$-norm estimate for the velocity field and an $O(h^k )$ convergence in $H^1$-norm estimate for the pressure. Finally, numerical experiments validate our theoretical results.