Finite element approximation of the Dirichlet problem using the boundary penalty method

Abstract

Summary. This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, A u =f, in a region f2cN" (n=2 or 3) by the boundary penalty method. If the finite element space defined over D h, a union of elements, has approximation power h ~ in the L 2 norm, then (i) for f2=D h convex polyhedral, we show that choosing the penalty parameter e=h x with 2>K yields optimal H 1 and L 2 error bounds if u~HK+I(Q);(ii) for Of 2 being smooth, an unfitted mesh (f2___D h) and assuming uEHK+2(f2) we improve on the error bounds given by Babuska [1]. As (ii)is not practical we analyse finally a fully practical piecewise linear approximation involving domain perturbation and numerical integration. We show that the choice 2=2 yields an optimal H 1 and interior L 2 rate of convergence for the error. A numerical example is presented confirming this analysis.

Model Adaptation Enriched with an Anisotropic Mesh Spacing for Nonlinear Equations: Application to Environmental and CFD Problems

“…The positiveness of σ and γ instills the logistic growth feature to the considered population, since the global reactive term −σu α +γu 2 α in (2.8) 1 can be rewritten in the standard form −σu α 1 − αγ/σu α , with σ the linear reproduction rate and σ/γ the carrying capacity of the environment [26,27]. The diffusive term in (2.8) models the random dispersion of the species; the advective term takes into account some possible transport phenomenon; f describes an external injection or withdrawal.…”

“…Critical issues to be tackled in the definition of the Lagrangian are the treatment of nonhomogeneous Dirichlet boundary data as well as the inclusion of possible stabilization terms in the discrete variational formulation. In particular, concerning the first issue, we resort to a penalty method (see, e.g., [1,9]). …”

“…In particular, we resort to the recipe provided in [1] for the case of convex polygonal domains, and choose ǫ(h) = ǫ 0 h 2 , where h = max K∈ h diam(K), and ǫ 0 ∈ + .…”

“…The penalty parameter ǫ = ǫ(h) is chosen according to [1]. We aim at monitoring the flow vorticity in the observation area, Ω O = (1, 3) × (0, 0.5), for different values of Re, i.e., we pick the goal functional as…”

A new fictitious domain approach for Stokes equation

“…As the generation of boundaryfitted meshes of good quality is a rather complex, often time consuming high costly, the fictitious domain method has been developed in order to overcome the meshing and re-meshing problem. One method for describing the essential Dirichlet boundary conditions, fitting into the context, is the Boundary penalty method in [1]. The real advantage of it is the combination with Nitsche's method in a fictitious domain in order to impose the essential boundary condition accurately in a weak sense.…”