Extrinsic Meshless Collocation Methods for PDEs on Manifolds

Abstract

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft. We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, i.e., Kansa methods. Our main theoretical contribution is the robust convergence analysis under some standard smoothness assumptions for high-order convergence. Numerical demonstrations are provided to verify the proven convergence rates, and we simulate reaction-diffusion equations for generating Turing patterns on manifolds. \bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs. kernel-based methods, Kansa methods, radial basis functions, convergence analysis \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs. 65D15, 65N35, 41A63 \bfD \bfO

Meshless Collocation Methods for Solving Pdes on Surfaces

“…for some sufficiently dense set of collocation points X ⊂ S. In [15], we show that the numerical solution U of (15) satisfies the estimate…”

Kernel-Based Meshless Collocation Methods for Solving Coupled Bulk–Surface Partial Differential Equations

“…We will solve the coupled problem in (1)-( 3) for the solutions (u B , u S ) by a domain-type meshless collocation method in [6] and an intrinsic surface-type approach in [4]. Our method is based on the coupled trial space U Z Ω ,Ω,Φ μ B × U Z S ,S,Ψ μ S in ( 9) and (10).…”

“…/2 and ν ≥ 3.5 are essential in terms of the H ν−3/2 (S) error only for the surface PDE (22), see [4]. Therefore, in this case, we need…”

“…In this work, we propose a bulk-surface meshless collocation method by combining the domain and surface-type approaches respectively in [4,6]. The latter method is intrinsic to surface without extension to narrow domains for embedded PDEs used in [5,[11][12][13]18,19].…”

Kernel-based collocation methods for heat transport on evolving surfaces

“…With the increasing numbers of both centers and collocation points, the L ∞ [0, T ]; L 2 (S(t)) errors decrease. We then compare results in Table 1 with those listed in Table 2 obtained by the corresponding exactly deter- (S(t)) errors and the corresponding estimated order of convergence (eoc) by (17) with respect to h Z , obtained by Algorithm 1 with overdetermined formulas (n X > n Z ), under the same other settings as in Figure 3. (S(t)) and L ∞ [0, T ]; L 2 (S(t)) errors and the corresponding convergence rates (eoc) with respect to h, obtained by Algorithm 1 with exactly determined formulas and the unfitted FEM [15], under the same other settings as in Figure 3.…”