SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON SURFACES WITH FUNDAMENTAL SOLUTIONS MENG CHEN * , KA CHUN CHEUNG † , AND LEEVAN LING ‡ Abstract. The aim of this paper is to present partial differential equations (PDEs) on surface to the community of methods of fundamental solutions (MFS). First, we present an embedding formulation to embed surface PDEs into a domain so that MFS can be applied after the PDEs is homogenized with a particular solution. Next, we discuss how the domain-MFS method can be used to directly collocate surface PDEs. Some numerical demonstrations were included to study the effect of basis functions and source point locations. 1. Partial differential equations on surfaces. In this paper, we focus on second-order elliptic partial differential equations (PDEs) posed on some sufficiently smooth, connected, and compact surface S⊂ R d with bounded geometry. Without loss of generality, we assume dim(S )= d − 1, a.k.a., S has co-dimension 1. We denote the unit outward normal vector at x ∈S as n = n(x) and the corresponding projection matrix to the tangent space of S at x as P (x)=[  P 1 ,...,  P d ](x) := I d − nn T ∈ R d×d , (1.1) where I d is the d × d identity matrix. Example 1. Let S be the unit circle in R 2 . Then, we have n =(x,y) T = x for every x ∈S and the projection matrix is P (x)=  1 − x 2 −xy −xy 1 − y 2  =  y 2 −xy −xy x 2  . Unlike standard PDEs posed in some bounded domains with flat geometry, cur- vatures of our computational domain S plays key roles in solution behaviours of the PDEs. The surface gradient ∇ S can be defined in terms of the standard Euclidean gradient ∇ for R d via projection P as ∇ S := P∇ (1.2) and the Laplace-Beltrami Δ S operators (a.k.a. the surface Laplacian) can then be * ASTRI Hong Kong Applied Science and Technology Research Institute Company Limited † NVIDIA AI Technology Center (NVAITC), NVIDIA, USA ‡ Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. 1