Approximate L² Error Control by Solution Post-Processing for Finite Element Solutions of PDEs with Higher-Order Adaptive Methods
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bottotornielli-mbottot-csesm-ccse-2025-thesis Marcos Botto Tornielli.pdf
Description
Thesis PDF
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13.76 MB
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Adobe PDF
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Author(s)
Botto Tornielli, Marcos Julian
Advisor(s)
Darmofal, David L.
Date Issued
September 2025
Publisher
Massachusetts Institute of Technology
Abstract
With the substantial computing resources available today, computational fluid dynamics simulations allow scientists and engineers to simulate physical problems very accurately. However, achieving this accuracy requires a sufficiently refined computational mesh, which is a primary driver for the high cost of complex simulations. Mesh adaptation methods provide an automated way to determine the regions where a mesh needs the most refinement and generate a new mesh that efficiently targets these regions. In this thesis, we build on previous work in a posteriori error estimation and mesh adaptation for finite element methods to propose a new mesh adaptation method based on L² error control by solution post-processing. A key feature of our method is its natural extension to higher-order discretizations while providing a problem-independent adaptation methodology. Problem-independent adaptation methods do not depend on specific information about the partial differential equation (PDE) problem being solved, and can therefore be applied to a wide range of problems without modification. We present numerical results applying the approximate L² error control method to a two-dimensional advection-diffusion problem with anisotropic features. These results demonstrate the proposed method’s ability to generate well-adapted anisotropic meshes for solutions with polynomial orders 1, 2, and 3. We also apply the approximate L² error control method to a more complex two-dimensional Reynolds-Averaged Navier-Stokes problem with turbulent flow over a flat plate. We compare the convergence of the drag coefficient and the characteristics of adapted meshes obtained with the proposed method and with an output-based adaptation approach. As expected, the approximate L² error control method is not as effective as the output-based approach in reaching a converged drag coefficient value, but it nevertheless demonstrates the ability to effectively control the approximate L² error in the Mach field.
MIT Department
Massachusetts Institute of Technology. Center for Computational Science and Engineering
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