Summary: Euler’s elastica digital inpainting model of T. F. Chan, S. H. Kang and J. Shen, introduced in [SIAM J. Appl. Math. 63, No. 2, 564–592 (2002; Zbl 1028.68185)], is well known for its attractive features of reconnecting contours along large distances and reconstructing the curvature of missing parts of objects and its ability to denoise outside the inpainting region. Since the underlying Euler-Lagrange partial differential equation is of fourth order and highly nonlinear, unfortunately, the usual numerical algorithm to find the solution is a very slow time-marching method (due to stability restriction). In this paper we address this fast solution issue by progressively proposing first a new unconditionally stable time-marching method and then a novel fixed-point method. The latter turns out to be two orders of magnitude faster than the time-marching method. Moreover, taking this new fixed-point method as a smoother, we develop an even faster nonlinear multigrid method for optimal performance. Numerical results are presented to illustrate the improved results obtained.