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Optimal convergence of a discontinuous-Galerkin-based immersed boundary method. (English) Zbl 1269.65108

Summary: We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier by A. J. Lew and G. C. Buscaglia [Int. J. Numer. Methods Eng. 76, No. 4, 427–454 (2008; Zbl 1195.76258)]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson’s problem with homogeneous boundary conditions over two-dimensional \(C{^2}\)-domains. For solutions in \(H{^q}\) for \(q >2\), we prove that the method constructed with polynomials of degree one on each element approximates the function and its gradient with optimal orders \(h{^2}\) and \(h\), respectively. When \(q\) = 2, we have \(h^{2-{\epsilon}}\) and \(h^{1-\epsilon}\) for any \(\epsilon >0\) instead. To this end, we construct a new interpolant that takes advantage of the discontinuities in the space, since standard interpolation estimates lead here to suboptimal approximation rates. The interpolation error estimate is based on proving an analog to Deny-Lions’ lemma for discontinuous interpolants on a patch formed by the reference elements of any element and its three face-sharing neighbors. Consistency errors arising due to differences between the exact and the approximate domains are treated using Hardy’s inequality together with more standard results on Sobolev functions.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Citations:

Zbl 1195.76258
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