A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Scientific Computing

Background:

• A posteriori error estimation is crucial for assessing the accuracy of numerical methods.
• The discontinuous Galerkin (DG) method offers flexibility in handling complex geometries and solution properties.
• Existing DG methods often rely on interior penalty approaches, motivating exploration of alternative formulations.

Purpose of the Study:

• To develop and present a novel a posteriori error approximation concept for the two-dimensional discontinuous Galerkin (DG) method.
• To utilize residuals and unique DG properties for effective error estimation.
• To demonstrate the applicability of the developed method on polygonal meshes and for adaptive hp-refinement.

Main Methods:

• The study employs a discontinuous Galerkin method with finite difference (DGFD) enforcement of solution continuity.
• Error approximation is constructed in an enriched space using hierarchical basis functions.
• Polygonal finite elements, including quadrilateral and triangular elements, are considered within the DG framework.

Main Results:

• Benchmark examples involving Poisson and linear elasticity problems were analyzed.
• Error estimation maps showed a strong correlation with exact errors across various mesh densities and approximation orders.
• The proposed error approximation concept was successfully applied to adaptive hp mesh refinement.

Conclusions:

• The presented a posteriori error approximation concept is simple, effective, and leverages DG method properties.
• The DGFD approach with polygonal elements provides a robust framework for error estimation.
• The method shows promise for adaptive mesh refinement strategies in complex simulations.