Immersed finite element methods for convection diffusion equations

Gwanghyun Jo; Do Y. Kwak; Gwanghyun Jo; Do Y. Kwak

doi:10.3934/math.2023407

AIMS Mathematics

2023, Volume 8, Issue 4: 8034-8059. doi: 10.3934/math.2023407

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Immersed finite element methods for convection diffusion equations

Gwanghyun Jo ¹,
Do Y. Kwak ^{2
,
,}

1.
Department of Mathematics, Kunsan National University, Republic of Korea
2.
Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea

Received: 07 November 2022 Revised: 09 January 2023 Accepted: 17 January 2023 Published: 31 January 2023
MSC : 65M08, 65M15, 65M60

In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.
- immersed finite element method,
- convection-diffusion problem,
- interface problem,
- control volume,
- upwinding scheme
Citation: Gwanghyun Jo, Do Y. Kwak. Immersed finite element methods for convection diffusion equations[J]. AIMS Mathematics, 2023, 8(4): 8034-8059. doi: 10.3934/math.2023407

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Abstract

In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.

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