Finite element approximation of the 𝑝-Laplacian

Abstract

Abstract. In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p € (1, oo), /, and g , find u such that -V • (\Vu\"-2Vu) = f iniîcR2, u = g on a«.The finite element approximation is defined over Í2* , a union of regular triangles, yielding a polygonal approximation to Q. For sufficiently regular solutions u , achievable for a subclass of data /, g , and Í2 , we prove optimal error bounds for this approximation in the norm Wl •Q(Q!1), q = p for p < 2 and q e [ 1, 2] for p > 2, under the additional assumption that Qh Ç £2. Numerical results demonstrating these bounds are also presented.

Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems

“…One wellknown difficulty when working with the natural energy norm is that the derived error estimates are not sharp. This drawback has been circumvented by Barrett and Liu [6] upon introducing a so-called quasinorm, thereby achieving optimal approximation results. The quasi-norm of the error between the exact solution u and the approximation solution, say u h , is a weighted L 2 -norm of the gradient ∇(u − u h ), where the weight depends on ∇u and ∇u h .…”

Computation of Minimum Energy Paths for Quasi-Linear Problems

“…Furthermore for large p, the nonlinear diffusion is enhanced also suggesting convergence may be difficult. This intuitive feel for how the convergence depends on p was proved by Barrett & Liu [3] (see also [6,17]) for finite-element approximations of the p-Laplacian problem…”

“…The discretised form of equation (2.3) can be solved using either nonlinear conjugate gradient methods or nonlinear least square methods such as the trust-region or Levenberg-Marquardt method; see Barrett & Liu [3] for an application of the Polak-Ribiére conjugate gradient method applied to the p-Laplacian and Coleman & Li [7] for a description of the trust-region method. To impose the boundary conditions, we solve concurrently the discretised form of (1.1) on the interior of the domain and the boundary conditions.…”

“…Under additional regularity assumptions, Barrett & Liu [3] were able to prove more optimal error bounds. In particular, provided that u ∈ W 3,1 (Ω) ∩ C 2,(2−p)/p (Ω) and p ∈ (1, 2), they were able to show that the error bound is O(h) and in the case p > 2 if u ∈ W 1,∞ (Ω) ∩ W 2,2 (Ω) then the second error bound in (2.6) still holds.…”

“…This appears to be a particularly tricky numerical problem since the usual approach to solving p-Laplacian type problems is to use nonlinear conjugate gradient methods due to the possible singular diffusion when 1 < p < 2; see for example [3]. These methods are only able to compute minima of the energy functional (1.2) and will miss the saddle-type solutions we are trying to locate.…”