A posteriori error estimation and adaptive meshes

Consider minus Laplacian u=f on a bounded polygonal domain, with homogeneous Dirichlet boundary data, f in L2, and a conforming piecewise linear Galerkin solution on a shape-regular simplicial mesh. The error norm is the L2 norm of its gradient. Inside an element, the strong residual is f plus the elementwise Laplacian of the approximation; for linear elements that Laplacian vanishes. Across interior faces, the outward-normal derivative contributions fail to cancel pointwise, leaving a flux jump.

A residual estimator combines squared element residuals weighted by element diameter squared and squared interior normal-flux jumps weighted by face diameter. Count each interior face once in the global estimator, or split it equally between adjacent local indicators. No Neumann boundary residual is included in this homogeneous Dirichlet model; different boundary conditions require their corresponding terms. For variable or discontinuous diffusion coefficients, both the energy norm and robust residual weights must be reconsidered rather than copied from the unit-coefficient Poisson case.

Under these assumptions, global reliability bounds the energy error by a mesh-independent constant times the estimator, with the constant depending on the domain and mesh shape regularity. It does not say that the constant is one, nor that each local indicator individually bounds the error on its own element. An elliptic error can be affected by residuals elsewhere, so a purely elementwise upper-bound interpretation is generally false. A rigorous proof uses Galerkin orthogonality and a stable quasi-interpolant to turn residuals into an energy dual estimate.

Efficiency runs in the other direction: an indicator is controlled by error on a neighboring patch plus unresolved variation in the data. The latter data oscillation is typically the norm of h times the difference between f and an appropriate local polynomial approximation. Omitting it can make an efficiency assertion false for rapidly varying data. Reliability shows that the estimator does not miss the error, while efficiency prevents excessive overestimation up to patch and data terms; neither identifies the exact error without constants.

An adaptive loop solves, estimates, marks and refines. A bulk marking rule selects elements whose squared indicators account for a prescribed fraction of the total. Refinement must maintain conformity and shape regularity, often by refining additional neighbors. Convergence and optimal work rates require more than computing this loop once: the estimator, refinement strategy, marking and approximation class must satisfy the hypotheses of the chosen adaptive theorem. A shrinking global indicator on a few experiments is useful evidence but not that theorem.

The estimator assumes the discrete Galerkin problem has been solved accurately enough. A large algebraic error can contaminate the residual and the Galerkin orthogonality used in its proof, so stopping criteria should balance solver error against estimated discretization error. For the explicit one-dimensional illustration below, interior faces are points and the jump contribution is defined as h_e times the squared derivative jump. This is the one-dimensional counterpart of the face weighting, not an unannounced replacement of a two-dimensional face integral.