We prove new a posteriori error estimates for surface finite element methods (SFEM). Surface FEM approximate solutions to PDE posed on surfaces. Prototypical examples are elliptic PDE involving the Laplace-Beltrami operator. Typically the surface is
approximated by a polyhedral or higher-order polynomial approximation. The resulting FEM exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEM require practical access to geometric information about the surface in order to computably bound the geometric error. It is thus advantageous to allow for maximum flexibility in representing surfaces in practical codes when proving a posteriori error estimates for SFEM. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on $C^2$ surfaces. Proofs of error estimates optimally reflecting the geometric error instead employ the closest point projection, which is defined using the signed distance function. Because the closest point projection is often unavailable or inconvenient to use computationally, a posteriori estimates using the signed distance function have notable practical limitations. We merge these two perspectives by assuming {it practical} access only to a general parametric representation of the surface, but using the distance function as a {it theoretical} tool. This allows us to derive sharper geometric estimators which exhibit improved experimentally observed decay rates when implemented in adaptive surface finite element algorithms.
Elliptic partial differential equations on surfaces play an essential role in geometry, relativity theory, phase transitions, materials science, image processing, and other applications. They are typically governed by the Laplace-Beltrami operator. W
e present and analyze approximations by Surface Finite Element Methods (SFEM) of the Laplace-Beltrami eigenvalue problem. As for SFEM for source problems, spectral approximation is challenged by two sources of errors: the geometric consistency error due to the approximation of the surface and the Galerkin error corresponding to finite element resolution of eigenfunctions. We show that these two error sources interact for eigenfunction approximations as for the source problem. The situation is different for eigenvalues, where a novel situation occurs for the geometric consistency error: The degree of the geometric error depends on the choice of interpolation points used to construct the approximate surface. Thus the geometric consistency term can sometimes be made to converge faster than in the eigenfunction case through a judicious choice of interpolation points.
Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework also relies e
xtensively on accurate computation of harmonic forms. In this work we study the convergence properties of adaptive finite element methods (AFEM) for computing harmonic forms. We show that a properly defined AFEM is contractive and achieves optimal convergence rate beginning from any initial conforming mesh. This result is contrasted with related AFEM convergence results for elliptic eigenvalue problems, where the initial mesh must be sufficiently fine in order for AFEM to achieve any provable convergence rate.
Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple and clustere
d eigenvalues was provided by Dai et. al. (2014), who proved convergence and optimality of AFEM for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (non-computable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. Gallistl (2015) used a similar tool in order to prove that a standard adaptive FEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or non-constant diffusion coefficients, however, the arguments of Dai et. al. and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this note we provide this missing key step, thus showing that standard adaptive FEM for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.
Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent wor
k of Arnold, Falk and Winther cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwells equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian.
Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global pollut
ion term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result.
We derive a posteriori error estimates in the $L_infty((0,T];L_infty(Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estim
ates for linear parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allow ing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.
