Numerical analysis of parabolic problems with dynamic boundary conditions

Kovács, Balázs; Lubich, Christian

doi:10.1093/imanum/drw015

Cited by 40 publications

(67 citation statements)

References 40 publications

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“…For the numerical solution of the above examples we consider a linear finite element method. In the following, from Elliott & Ranner (2013) and (Kovács & Lubich, 2017, Section 3.2.1), we will briefly recall the construction of the discrete domain, the finite element space and the lift operation which can be used discretize the particular problems of Section 2.2 in space. Then we will present the abstract framework for spatial discretizations of (2.1) and state the main abstract error estimate.…”

Section: Spatial Discretization With the Finite Element Methodsmentioning

confidence: 99%

“…Error estimates for the Ritz map. From (Kovács & Lubich, 2017, Lemma 3.13 and 3.15) we recall the following estimates for the error of the Ritz map. Note the weaker norm on Γ for β = 0 due to a lack of boundary regularity of solutions of the Poisson equation with Neumann boundary conditions.…”

Section: 22mentioning

confidence: 99%

“…Using B ∈ L (V,V * ) and by the · norm error estimate for the Ritz map (Kovács & Lubich, 2017, Lemma 3.1) we find that foru ∈ H 2 (Ω ) ∩V…”

Section: )mentioning

confidence: 99%

“…In this section, we present an abstract setting for wave equations, similar to the ones in Kovács & Lubich (2017) and Hipp et al (2018), and then consider different examples of wave equations with dynamic boundary conditions fitting into this abstract framework.…”

Section: Analysis Of Wave Equations With Dynamic Boundary Conditionsmentioning

confidence: 99%

“…Note that, for example, the bilinear form a contains boundary terms which influence R h . Using the notation from (Kovács & Lubich, 2017, Section 3.4), we will write R h u := ( R h u) ∈ V h for the lifted Ritz map. In Section 5, we prove the following error estimate for the lifted solution of the semi-discrete abstract wave equations in the H-norm…”

Section: Of 44mentioning

confidence: 99%

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Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

Hipp

Kovács

2020

IMA Journal of Numerical Analysis

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L 2 norm error estimates of semi-and full discretisations, using bulk-surface finite elements and Runge-Kutta methods, of wave equations with dynamic boundary conditions are studied. The analysis resides on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed which fit into the abstract framework. For problems with velocity terms, or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than two. These can also be observed in the presented numerical experiments.

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Section: Spatial Discretization With the Finite Element Methodsmentioning

confidence: 99%

Section: 22mentioning

confidence: 99%

“…Using B ∈ L (V,V * ) and by the · norm error estimate for the Ritz map (Kovács & Lubich, 2017, Lemma 3.1) we find that foru ∈ H 2 (Ω ) ∩V…”

Section: )mentioning

confidence: 99%

Section: Analysis Of Wave Equations With Dynamic Boundary Conditionsmentioning

confidence: 99%

Section: Of 44mentioning

confidence: 99%

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Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

Hipp

Kovács

2020

IMA Journal of Numerical Analysis

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Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

Frittelli

Madzvamuse

Sgura

2023

Numerical Methods Partial

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In this work we present a novel bulk‐surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk‐surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. We present a geometric error analysis of bulk‐surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second‐order convergence in space, provided the exact solution is in the bulk and on the surface, where the additional is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. Two numerical examples on the unit sphere and on the Dupin ring cyclide confirm the convergence result.

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Bulk-surface virtual element method for systems of PDEs in two-space dimensions

2021

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In this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method (Beirão Da Veiga et al. in Math Models Methods Appl Sci 23(01):199–214, 2013. https://doi.org/10.1051/m2an/2013138) in the bulk domain to a surface finite element method (Dziuk and Elliott in Acta Numer 22:289–396, 2013. https://doi.org/10.1017/s0962492913000056) on the surface. The proposed method, which we coin the bulk-surface virtual element method includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes (Madzvamuse and Chung in Finite Elem Anal Des 108:9–21, 2016. https://doi.org/10.1016/j.finel.2015.09.002). The method exhibits second-order convergence in space, provided the exact solution is $$H^{2+1/4}$$ H 2 + 1 / 4 in the bulk and $$H^2$$ H 2 on the surface, where the additional $$\frac{1}{4}$$ 1 4 is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an $$L^2$$ L 2 -preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator (Elliott and Ranner in IMA J Num Anal 33(2):377–402, 2013. https://doi.org/10.1093/imanum/drs022) for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes (Madzvamuse and Chung 2016). Three numerical examples illustrate our findings.

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Numerical analysis of parabolic problems with dynamic boundary conditions

Cited by 40 publications

References 40 publications

Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

Bulk-surface virtual element method for systems of PDEs in two-space dimensions

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