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

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.

### 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

ANGENER; BL2D-V2
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