A posteriori error estimates for hypersingular integral equation on spheres with spherical splines

TL;DR

This work develops two a posteriori error estimation strategies for the hypersingular integral equation on the unit sphere solved by Galerkin methods with spherical splines: a residual-based bound and a hierarchical bound under a saturation assumption. The residual estimator relates the error to local residuals $\eta_{\Delta,s}(\tau)$ via $\|u-u_\Delta\|_{H^s(\mathbb S)} \le C\eta_{\Delta,s}(\mathbb S)$, while the hierarchical estimator uses refined meshes and nodal estimators $\mu_{\mathbf v}$ to bound the $H^{1/2}$-error by a sum of squared estimators. These results underpin adaptive mesh refinement that reduces degrees of freedom and computation time while preserving accuracy, as demonstrated in numerical experiments on exterior Neumann problems where adaptive methods outperform uniform refinement by large margins. The methods are framed on the sphere but are extendable to sphere-like geometries, enhancing practical applicability in geophysical and related simulations where data are gathered on spherical surfaces. Overall, the paper advances reliable, efficient adaptive schemes for hypersingular problems on spherical domains.

Abstract

A posteriori residual and hierarchical upper bounds for the error estimates were proved when solving the hypersingular integral equation on the unit sphere by using the Galerkin method with spherical splines. Based on these a posteriori error estimates, adaptive mesh refining procedures are used to reduce complexity and computational cost of the discrete problems. Numerical experiments illustrate our theoretical results.

Figures (10)

• Figure 1: Possible cases of refined triangles
• Figure 2: Uniform mesh refinement
• Figure 3: Errors vs DoFs for $f_1$
• Figure 4: Errors vs Accumulating computation time for $f_1$
• Figure 5: Errors vs DoFs for $f_2$

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof