Gradient Weighted Embedded Error Estimator for Mesh Adaptation

Mesh adaptation is essential for accurate computational fluid dynamics (CFD) simulations, especially when computational resources are limited.In simulating the advection-diffusion equation using the finite volume method (FVM), mesh adaptation is often necessary to accurately capture sharp gradients and complex flow features.Although the metric-based mesh adaptation used in this study is highly effective with true errors, these are typically unavailable, requiring the use of error estimators.However, adjoint-based error estimators, though effective, are computationally expensive as they require solving the larger adjoint system.To address this, an inexpensive error estimator based on the gradient of the numerical solution and the embedded method is proposed.Specifically, the central difference and upwind difference schemes are utilized for error estimation within this framework.Since errors are predominantly influenced by sharp gradients in the scalar boundary layer, a target functional based on the product of the embedded error and the solution gradient to compute the error estimator is proposed.Consequently, these proposed error estimators generate meshes closely resembling those produced using true error values, effectively resolving the scalar boundary layer.As a result, the L 2 and L norms of the error typically decrease with successive adaptation cycles.Furthermore, the L 1 , L 2 , and L norms of error per element also reduce with successive adaptation cycles.In conclusion, the proposed error estimator facilitates mesh adaptation without needing true errors or computationally expensive adjoint-based error estimators, thus ensuring accurate numerical solutions, especially in scenarios with sharp boundary layers.