Search

A new class of parametric finite element methods, with a new type of artificial tangential velocity constructed at the continuous level, is proposed for solving surface evolution under geometric flows. The method is constructed by coupling the normal velocity of the geometric flow with an artificial tangential velocity determined by a harmonic map from a fixed reference surface\scrM to the unknown surface\Gamma (t), formulated at the continuous level as a system of geometric partial differential equations in terms of a Lagrange multiplier. Since the harmonic map is almost anglepreserving, the new method could preserve the mesh quality, i.e., the shapes of the triangles, as long as the mesh quality of the reference surface is good. Extensive numerical experiments and benchmark examples are presented to demonstrate the convergence of the proposed method and the advantages of the method in preserving the mesh quality of the surfaces for mean curvature flow and surface diffusion, in comparison with other available methods such as the parametric finite element methods proposed by Barrett, Garcke, and N\"urnberg in 2008 and the DeTurck flow techniques proposed by Elliott and Fritz in 2017. [ABSTRACT FROM AUTHOR]

In this paper we construct a quadrature scheme to numerically solve the nonlocal diffusion equation (Aα + bI)u = f with Aα the αth power of the regularly accretive operator A. Rigorous error analysis is carried out and sharp error bounds (up to some negligible constants) are obtained. The error estimates include a wide range of cases in which the regularity index and spectral angle of A, the smoothness of f, and the size of b and α are all involved. The quadrature scheme is exponentially convergent with respect to the step size and is root-exponentially convergent with respect to the number of solves. Some numerical tests are presented in the last section to verify the sharpness of our estimates. Furthermore, both the scheme and the error bounds can be utilized directly to solve and analyze time-dependent problems. [ABSTRACT FROM AUTHOR]