Abstract:
Errors and uncertainties in finite element method (FEM) computing can come from eight sources: four FEM-method-specific and four model-specific. By treating every FEM solution as a numerical experiment for a fixed model, solution verification can be achieved by quantifying errors and uncertainties from the first four sources, then developing algorithms and metrics to assess the accuracy of candidate solutions. This paper introduces a new approach to FEM verification using three mathematical methods and formulating three metrics for solution accuracy assessment. These methods include a 4-parameter logistic function for asymptotic solutions, nonlinear least squares for confidence bounds, and the Jacobian definition for element Jacobians. These tools estimate FEM solution uncertainty at large degrees of freedom (d.o.f.), gain in percent relative error (PRE) rate per d.o.f., and the estimated mean of the Jacobian distribution. These quantities serve as metrics for assessing solution accuracy to achieve a "best" estimate with uncertainty quantification. The results include metric calibration using problems with known analytical solutions and application to problems without known theoretical solutions.
