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

TL;DR

The paper tackles non-homogeneous Neumann boundary conditions on curved domains by applying a boundary value correction that shifts the problem from the true boundary $\Gamma$ to a surrogate $\Gamma_h$ and uses a Taylor-expansion-based transfer of flux data. It analyzes a high-order Raviart-Thomas mixed finite element method on $\Omega_h$, proving convergence results: an $O(h^{k+1/2})$ rate in $L^2$ for the velocity and an $O(h^k)$ rate in $H^1$ for the pressure, under a geometry error bound $\delta_h\lesssim h^2$. The work also demonstrates, both theoretically and numerically, that neglecting boundary correction leads to suboptimal velocity accuracy on curved boundaries, highlighting the method’s necessity and efficacy. The approach avoids cut elements, simplifies implementation, and provides a rigorous treatment of geometry-induced approximation loss, with practical relevance for high-order simulations on curved domains.

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.

Figures (1)

• Figure 1: (a). The true boundary $\Gamma$ (blue curve), the approximated boundary $\Gamma_h$ (red lines) and the typical region $\cup_{e\in\mathcal{E}_{h}^b}\Omega_h^{e}$ bounded by $\Gamma$ and $\Gamma_h$. (b). The distance $\delta_h(\boldsymbol x_h)$ and the unit vector $\boldsymbol{\nu_h}$ to $\Gamma_h$.

Theorems & Definitions (31)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 4.1
• Lemma 4.2