Numerical algorithms of subdivision-based IGA-EIEQ method for the molecular beam epitaxial growth models on complex surfaces

In this paper, we develop fully discrete methods for two molecular beam epitaxial models on surfaces, where one has a fourth-order Ginzburg-Landau double-well potential, and the other has a logarithmic nonlinear potential. The main challenge is how to establish high-order spatiotemporal accurate schemes with the property of unconditional energy stability. The spatial numerical approach adopted in this paper is the recently developed isogeometric analysis framework based on loop subdivision technology, where it can provide an exact and elegant description for the surfaces with arbitrary topology, and subdivision basis functions can serve to represent the numerical solutions. The temporal discrete scheme is based on the "explicit-invariant energy quadratization" strategy. Its novelty is to introduce two auxiliary variables (one local and the other nonlocal) and then develop two specific ordinary differential equations that convert the original equations into their equivalent forms. The use of the nonlocal variable helps us avoid dealing with variable coefficients and solve only equations with constant coefficients, and the use of the local variable allows us to obtain linear schemes with energy-stable structures numerically. We further perform various numerical experiments on some well-known complex surfaces, such as the sphere, bubble, and splayed surface to verify the stability and accuracy of the developed method.