Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations
In this paper, we are interested in studying a mixed formulation of weak Galerkin type to approach the electric field and a Lagrange multiplier, which are solutions of a problem deriving from Maxwell’s equations. Our numerical scheme is formed with stable finite elements constructed of usual polynom...
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2024-12-01
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| author | Abdelhamid Zaghdani Abdelhalim Hasnaoui |
| author_facet | Abdelhamid Zaghdani Abdelhalim Hasnaoui |
| author_sort | Abdelhamid Zaghdani |
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| description | In this paper, we are interested in studying a mixed formulation of weak Galerkin type to approach the electric field and a Lagrange multiplier, which are solutions of a problem deriving from Maxwell’s equations. Our numerical scheme is formed with stable finite elements constructed of usual polynomials of degree <i>k</i> for the electric field and of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> for the Lagrange multiplier; its consistency and well-posedness are shown. Some optimal error estimates are proven and tested numerically in a bounded subdomain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>2</mn></msup></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-e95d2813f04047a4bbdcd47b059789c7 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-e95d2813f04047a4bbdcd47b059789c72024-12-27T14:37:57ZengMDPI AGMathematics2227-73902024-12-011224390110.3390/math12243901Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s EquationsAbdelhamid Zaghdani0Abdelhalim Hasnaoui1Faculty of Arts and Science, Northern Border University, Rafha P.O. Box 840, Saudi ArabiaFaculty of Arts and Science, Northern Border University, Rafha P.O. Box 840, Saudi ArabiaIn this paper, we are interested in studying a mixed formulation of weak Galerkin type to approach the electric field and a Lagrange multiplier, which are solutions of a problem deriving from Maxwell’s equations. Our numerical scheme is formed with stable finite elements constructed of usual polynomials of degree <i>k</i> for the electric field and of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> for the Lagrange multiplier; its consistency and well-posedness are shown. Some optimal error estimates are proven and tested numerically in a bounded subdomain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>2</mn></msup></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/12/24/3901weak Galerkin methodsmixed finite element methodsthe Maxwell equations |
| spellingShingle | Abdelhamid Zaghdani Abdelhalim Hasnaoui Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations Mathematics weak Galerkin methods mixed finite element methods the Maxwell equations |
| title | Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations |
| title_full | Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations |
| title_fullStr | Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations |
| title_full_unstemmed | Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations |
| title_short | Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations |
| title_sort | analysis of a weak galerkin mixed formulation for modified maxwell s equations |
| topic | weak Galerkin methods mixed finite element methods the Maxwell equations |
| url | https://www.mdpi.com/2227-7390/12/24/3901 |
| work_keys_str_mv | AT abdelhamidzaghdani analysisofaweakgalerkinmixedformulationformodifiedmaxwellsequations AT abdelhalimhasnaoui analysisofaweakgalerkinmixedformulationformodifiedmaxwellsequations |