Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Appro
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Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Approximations Shun Zhang1 Received: 11 July 2019 / Revised: 27 March 2020 / Accepted: 15 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the Raviart–Thomas and Brezzi– Douglas–Marini mixed finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect to the local regularity of the solution. Several versions of the robust best approximations of the flux and the potential approximations are obtained. These robust and local optimal a priori estimates provide guidance for constructing robust a posteriori error estimates and adaptive methods for the mixed approximations. Keywords Mixed finite element method · Robust a priori error estimate · Local optimal error estimate · Low regularity · Interface problems
1 Introduction As a prototype of problems with interface singularities, this paper studies a priori error estimates of mixed finite element methods for the following interface problem (i.e., the diffusion problem with discontinuous coefficients): − ∇ · (α(x)∇ u) = f in Ω
(1.1)
with homogeneous Dirichlet boundary conditions (for simplicity) u = 0 on ∂Ω, Rd
(1.2)
where Ω is a bounded polygonal domain in with d = 2 or 3; f ∈ is a given function; and diffusion coefficient α(x) is positive and piecewise constant with possible large jumps across subdomain boundaries (interfaces): L 2 (Ω)
This work was supported in part by Hong Kong Research Grants Council under the GRF Grant Project No. CityU 11305319.
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Shun Zhang [email protected] Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China 0123456789().: V,-vol
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Journal of Scientific Computing
(2020) 84:40
α(x) = αi > 0 in Ωi for i = 1, . . . , n. n is a partition of the domain Ω with Ωi being an open polygonal domain. It is Here, {Ωi }i=1 well known that the solution u of problem (1.1) belongs to H 1+s (Ω) with possibly very small s > 0, see for example Kellogg [23]. But we should also note that even the global regularity is low, when a finite element mesh is given, the singularity or those elements whose solution having a large gradient often only appear near some points, or along a curve. Thus it is not optimal to use the global regularity and a global uniform mesh-size to do the a priori error estimate. In [10], we introduced the idea of robust and local optimal a priori error estimate. The robustness means that the genetic constants appeared in the estimates are independent of the parameters of the equation, the coefficient α in our case. The local optimality means that in the error estimate, the upper bound is optimal with the regularity of each element and local mesh sizes, instead of using a global uniform mesh size and a globa
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