{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/164837"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/164837","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Approximate L² Error Control by Solution Post-Processing for Finite Element Solutions of PDEs with Higher-Order Adaptive Methods","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Botto Tornielli, Marcos Julian"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["Approximate L² Error Control by Solution Post-Processing for Finite Element Solutions of PDEs with Higher-Order Adaptive Methods"]}]}],"canonical_facts":{"dc:contributor.advisor":["Darmofal, David L."],"dc:contributor.department":["Massachusetts Institute of Technology. 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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."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/164837"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"dc:rights.uri":["https://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Approximate L² Error Control by Solution Post-Processing for Finite Element Solutions of PDEs with Higher-Order Adaptive Methods"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Computational Science and Engineering"]},"updated_at":"2026-07-22T22:22:02Z"}