This paper aims to propose a novel yet simple approach to the adaptive finite element (FE) analysis of two-dimensional boundary value problems (BVPs) in the maximum norm with local refinement of unstructured meshes based on the reduced element technique.
The main idea of the reduced element technique involves strategically decomposing the conventional polynomial finite element (FE) solution on each element into two components: a reduced solution of one degree lower, and a built-in pointwise error estimator provided by the highest-degree terms. The convergence order discrepancy between the original FE solution and the reduced solution provides theoretical justification for the effectiveness of this error estimator. Consequently, an adaptive analysis algorithm can be developed, with the reduced solution serving as the final validated result.
Representative numerical examples, including Poisson's equation, plane elasticity, the Mindlin–Reissner plate bending problem and free vibration analysis of elastic membranes, were analyzed; the results show that the final adapted meshes reasonably reflect the local difficulties inherent in the physical problems and the proposed adaptive analysis can produce reduced solutions that satisfy the user-specified tolerances in the maximum norm at nearly optimal adaptive convergence rates.
The proposed reduced element technique integrates error estimation and solution validation at the element level within a single adaptive procedure. It ensures that the adapted reduced solutions satisfy specified maximum-norm error tolerances at nearly optimal convergence rates. Its simplicity and versatility make it extensible to other challenging problems.
