An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

Abstract

Abstract. We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.A posteriori error estimates are given in the energy norm and the L 2 -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

“…We assume throughout the paper that Ω is a convex polygonal domain, noting that our results could be extended to cover certain nonconvex domains, such as domains with reentrant corners in d = 2, by using weighted a posteriori estimates for elliptic problems [22]. Since the difficulties in the analysis below in the case of other boundaries are mainly coming from the elliptic part of the error, the reader interested in a posteriori error estimates for curved boundaries is referred to Dörfler and Rumpf [12]. We will consider the problem of finding a finite element approximation of the solution u ∈ L ∞ (0, T ; H…”

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

“…In particular, we do not consider problems with nonconstant coefficients, the case where a(Γ h ) = Γ, lower-order terms, or nonhomogeneous Dirichlet or Neumann boundary conditions. These additional complexities may be handled in much the same way as for problems on polygonal domains in R 2 , so we refer for example to the works [DR98], [DW00], [BCD04], [MN05], and [AO00] where many of these issues are considered. Under suitable assumptions, our development also holds largely unchanged for surfaces of codimension 1 which are immersed in R n , n ≥ 2.…”

Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems

“…However, the proper treatment of finite element approximations involving curved boundaries is somewhat technical even when considering a posteriori energy-norm bounds (cf. [DR98]), and we do not wish to clutter our presentation. Second, the constants in our a posteriori estimates depend on the unknown solution u in nonlinear problems and even in linear problems may depend on the coefficients in a fashion that will require a local weighting of the residuals.…”