Anisotropic \(hp\)-adaptive method based on interpolation error estimates in the \(H^1\)-seminorm. (English) Zbl 1363.65208

Dolejší, Vít

Appl. Math., Praha 60, No. 6, 597-616 (2015).

The construction of anisotropic \(hp\)-meshes \({\mathcal T}_{hp} = \{{\mathcal T}_h,\pmb{p}\}\) is discussed, where \({\mathcal T}_h = \{K\}\) is a triangulation of a given domain \(\Omega\) and \(\pmb{p} = \{p_K ; K\in {\mathcal T}_h\}\) is a set of polynomial degrees with \(p_K > 0\). The following problem is considered: Let be given a function \({u \in V = C^\infty(\Omega)}\). Find an \(hp\)-mesh \({\mathcal T}_{hp}\) such that (i) \({| u - \Pi_{hp}u|_{H^1({\mathcal T}_h)} \leq \omega}\), where \(\omega\) is a given tolerance and \(\Pi_{hp}\) is an interpolation operator from \(V\) in the space of discontinuous piecewise polynomial functions, (ii) the number of degrees of freedom, i.e. the dimension of \(S_{hp}\), is minimal. Instead of this problem auxiliary local problems are solved, which results in an \(hp\)-mesh which is close to the solution of the problem formulated above. A corresponding algorithm for the construction of an anisotropic \(hp\)-mesh is given. Furthermore it is discussed how the presented approach can be extended to the mesh optimization with respect to the broken \(W^{k,q}\)-seminorm, where \(k \geq 1\) and \(q \in [1,\infty)\). It is also explained how the proposed mesh generation algorithm can be applied to the numerical solution of second order boundary value problems (b.v.p.’s). Hereby, the function \(u\) is replaced by the approximate solution \(u_{hp} \in S_{hp}\) of the b.v.p. and the algorithm is applied iteratively, i.e. one constructs a mesh, solves the b.v.p. approximately, constructs a new improved mesh and so on. Finally, numerical examples are given to demonstrate the efficiency of the proposed method. Hereby, a linear convection-diffusion problem with boundary layers and with double curved interior layers is considered. The new algorithm is compared with the application of \(hp\)-isotropic meshes, \(h\)-anisotropic meshes, where the polynomial degree is fixed, and with \(hp\)-anisotropic \(L^2\)-optimal meshes.

Reviewer: Michael Jung (Dresden)

MSC:

65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Keywords:

\(hp\)-methods; anisotropic mesh adaptation; interpolation error estimates; convection-diffusion problems; boundary layers; interior layers; ANGENER; BL2D-V2; algorithm; mesh optimization; second order boundary value problems; numerical examples

Software:

ANGENER; BL2D-V2

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