Residual error estimators and reliability
Bound the unknown energy error from element residuals and flux jumps while retaining data oscillation and patch dependence.
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The question for today
What you’ll understand
- Construct the residual estimator for homogeneous Poisson with conforming P1 elements.
- Distinguish global reliability from local efficiency and their constants.
- Explain the role of data oscillation and the absence of a global H² assumption.
An a priori estimate predicts error from assumed smoothness, while an a posteriori estimate uses the computed solution and data to control the error afterward. For Poisson’s equation the residual splits naturally into source imbalance inside elements and normal-flux jumps across their faces. Multiplying these terms by the right local length scales produces an estimator of the gradient error. The estimator is a computable bound under hypotheses, not an exact error readout or a universal certificate for every diffusion operator.
Integrate the residual element by element
Take a bounded polygonal domain in two dimensions, homogeneous full Dirichlet data, source f in L², and the conforming piecewise affine Galerkin solution on a fitted shape-regular triangulation. Inside each triangle its Laplacian is zero, so the element residual f plus the discrete Laplacian is simply f. Across an interior edge, add the two outward normal derivatives of u_h to define the flux jump. A globally smooth exact gradient has canceling normal fluxes there.
The standard squared estimator sums h_K squared times the squared L² element residual and h_E times the squared L² edge jump, counting each interior edge once. Boundary terms would change for Neumann data, and variable diffusion requires the physical flux rather than an unweighted gradient jump. Our model has diffusion exactly one, so no unresolved coefficient-contrast claim is hidden in the constant. The formulas also assume the discrete variational problem has been solved accurately enough for Galerkin orthogonality to be used.
Reliability and efficiency point in opposite directions
Global reliability bounds the true gradient error by a mesh-independent constant times the estimator. A proof tests the residual against the error minus a stable local quasi-interpolant, then uses Galerkin orthogonality, patch approximation and trace estimates. This argument does not require a global H² solution or convexity-based elliptic regularity. Its role is to guarantee that unresolved error cannot remain large while the computable residual estimator is small in the stated setting.
Local efficiency bounds an element indicator by the error on a neighboring patch plus a data oscillation term. Oscillation measures source components not captured by a local polynomial projection and cannot generally be omitted for arbitrary L² data. The patch is substantive: an indicator cannot in general certify only the error inside that same cell. If a projected source is used in the computable residual, a corresponding oscillation contribution is also needed in reliability rather than silently treating the projected source as exact.
Interpret constants and computational error honestly
The constants depend on the domain, polynomial degree and mesh shape regularity, not on local cell size within the controlled family. Their existence is not a statement that the estimator equals the error or has constant one. A large estimator can be driven partly by data oscillation rather than by a proportionally large solution error. Combining reliability and efficiency therefore gives information about both approximation and data resolution, which becomes important in adaptive refinement analysis.
A residual estimator derived for the exact discrete solution may need an additional algebraic-error contribution when the linear system is solved inexactly. Approximate boundary data, curved-domain geometry, and coefficient discontinuities likewise require their own terms or modified estimates. These limits do not make the estimator unhelpful; they identify the problem for which its guarantee is valid. Separating global upper control, local lower control, and modeling or solver error prevents an attractive residual plot from being mistaken for an unconditional proof of accuracy.
Work it through
A square with no interior P1 degree of freedom
Triangulate the unit square by one diagonal and impose zero values at all four vertices. Let u=x(1−x)y(1−y), so f=2[x(1−x)+y(1−y)]. Compute u_h, its gradient error and the residual estimator using each triangle’s diameter √2.
Every vertex lies on the Dirichlet boundary, so the conforming P1 space has no nonzero free function and u_h=0. All flux jumps vanish. Direct integration gives squared gradient error 2·(1/3)·(1/30)=1/45.
Writing q(t)=t(1−t), its integrals are ∫q=1/6 and ∫q²=1/30. Hence the squared L² norm of f is 4[2/30+2/36]=22/45.
Each triangle has diameter squared two, so the volume estimator sum is twice the global f norm squared: eta_h²=44/45. Thus eta_h divided by the true gradient error is √44, not one.
The estimator is computable without knowing u, while this manufactured solution lets us audit it exactly. One example does not establish a universal reliability constant.
The living notebook
Explore the model
Interactive lab · function
A small candidate curve can leave a nonzero residual
On paper define the zero-endpoint candidate u_a=a(x²−x) for −u″=2. Reset and set a=0,x=0.5; predict zero quadratic value and residual 2+2a=2. Reset, set a=−1,x=0.5; the quadratic contribution is −0.25, and adding −a x gives candidate value 0.25 with residual zero. The instrument supplies only a x² and its derivatives; residual and affine correction are manual calculations, not an estimator implementation. Changes retain time; Reset restores both settings and zero time.
- y at chosen x
- 0.608
- Tangent slope · 2ax
- 1.35
- Signed area · ax³/3
- 0.182
This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forFunction size alone does not bound PDE error; the source and differential residual must be included.
Pause & reason
Check your understanding
Your turn
Try it yourself
Suppose a global estimator is 0.04 with a verified reliability constant two. A local indicator is 0.03, its efficiency constant is three, and its patch oscillation is 0.002. What global upper bound and patch-error lower bound follow?
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Use reliability in its stated direction. Rearrange eta_K≤C_eff(error_patch+osc_patch) for the local lower bound.
Compare with the solution
Global reliability gives gradient error at most two times 0.04, or 0.08. Local efficiency gives patch error at least 0.03/3−0.002=0.008. This is a patch lower bound, not a same-cell error value. If the rearranged expression were negative, only the trivial nonnegative lower bound would follow. Neither estimate asserts that the true error equals the computed indicator.
Keep the idea
Take it with you
- Element residuals and normal-flux jumps combine with local length weights.
- Reliability is global; local efficiency uses neighboring patches and data oscillation.
- The guarantee is conditional on the stated PDE, mesh, boundary data and algebraic accuracy assumptions.
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