An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations
We construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discont...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Mathematical and Computational Applications |
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| Online Access: | https://www.mdpi.com/2297-8747/28/1/7 |
| _version_ | 1797619434984046592 |
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| author | Juan F. Giraldo Victor M. Calo |
| author_facet | Juan F. Giraldo Victor M. Calo |
| author_sort | Juan F. Giraldo |
| collection | DOAJ |
| description | We construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discontinuous Galerkin formulation in space. This combination delivers a stable continuous solution and an on-the-fly error estimate that robustly guides adaptivity at every discrete time. We show the performance of advection-dominated problems to demonstrate stability in the solution and efficiency in the adaptivity strategy. We also present the method’s robustness in the nonlinear Bratu equation in two dimensions. |
| first_indexed | 2024-03-11T08:28:08Z |
| format | Article |
| id | doaj.art-30b1dcef16a248c48991783b933d94d5 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-11T08:28:08Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-30b1dcef16a248c48991783b933d94d52023-11-16T21:57:58ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472023-01-01281710.3390/mca28010007An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction EquationsJuan F. Giraldo0Victor M. Calo1School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, AustraliaSchool of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, AustraliaWe construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discontinuous Galerkin formulation in space. This combination delivers a stable continuous solution and an on-the-fly error estimate that robustly guides adaptivity at every discrete time. We show the performance of advection-dominated problems to demonstrate stability in the solution and efficiency in the adaptivity strategy. We also present the method’s robustness in the nonlinear Bratu equation in two dimensions.https://www.mdpi.com/2297-8747/28/1/7residual minimizationunsteady advection–diffusion equationsdiscontinuous Galerkinimplicit time-marching schemesadaptive mesh refinement |
| spellingShingle | Juan F. Giraldo Victor M. Calo An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations Mathematical and Computational Applications residual minimization unsteady advection–diffusion equations discontinuous Galerkin implicit time-marching schemes adaptive mesh refinement |
| title | An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations |
| title_full | An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations |
| title_fullStr | An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations |
| title_full_unstemmed | An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations |
| title_short | An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations |
| title_sort | adaptive in space stabilized finite element method via residual minimization for linear and nonlinear unsteady advection diffusion reaction equations |
| topic | residual minimization unsteady advection–diffusion equations discontinuous Galerkin implicit time-marching schemes adaptive mesh refinement |
| url | https://www.mdpi.com/2297-8747/28/1/7 |
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