An Adaptive Finite Element Method for the Laplace–Beltrami Operator on Implicitly Defined Surfaces

We present an adaptive finite element method for approximating solutions to the Laplace–Beltrami equation on surfaces in R³ which may be implicitly represented as level sets of smooth functions. Residual‐type a posteriori error bounds which show that the error may be split into a “residual part” and a “geometric part” are established. In addition, implementation issues are discussed and several computational examples are given.