Volume 9, Issue 1

We present an efficient adaptive method for unstructured triangular meshes especially developed for the numerical simulation of viscous compressible flow. Thismethod uses the anisotropic mesh adaptation (AMA) technique introduced in [Dolejši, Computing and Visualization in Science 1: 165–178, 1998, Dolejši and Felcman, Numer. Meth. Part. Diff. Equations] with some additional improvement reflecting physical properties of viscous high speed flow. We summarize the AMA method with its geometrical interpretation and we describe the numerical solver, which uses the combination of finite volume and finite element methods. We show that the direct application of AMA technique is not suitable for the viscous flow simulation and we propose some modifications, namely for capturing boundary layers and a wake. Moreover an additional smoothing technique increasing the regularity of the triangulation is presented.

We develop a data-sparse and accurate approximation of the normalised hyperbolic operator sine family generated by a strongly P-positive elliptic operator defined in [Gavrilyuk, J. Math. Anal. Appl. 236: 327–349, 1999, Gavrilyuk and Makarov, Numer. Funct. Anal. Optimization 20: 695–717, 1999]. In the preceding papers [Hackbusch, Computing 62: 89–108, 1999–Hackbusch, Khoromskij, and Sauter, On ℋ 2 -matrices, Springer-Verlag, 2000], a class of ℋ-matrices has been analysed which is data-sparse and allow an approximate matrix arithmetic with almost linear complexity. An ℋ-matrix approximation to the operator exponent with a strongly P-positive operator was proposed in [Gavrilyuk, Hackbusch, and Khoromskij, ℋ-matrix approximation for the operator exponential with applications, 2000]. In the present paper, we apply the ℋ-matrix techniques to approximate the elliptic solution operator on cylinder-like domains Ω × [ a, b ] associated with the elliptic operator , x ∈ [ a, b ]. It is explicitly presented by the normalised hyperbolic sine of an elliptic operator ℒ defined in Ω. The approach is then applied to elliptic partial differential equations in domains composed of rectangles or cylinders. In particular, we consider the ℋ-matrix approximation to the interface Poincaré-Steklov operators with application in the Schur-complement domain decomposition method. Starting with the Dunford–Cauchy representation for the hyperbolic sine operator, we then discretise the integral by the exponentially convergent quadrature rule involving a short sum of resolvents. The latter are approximated by the ℋ-matrix techniques. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [ a, b ].

On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has two exponential boundary layers. We apply the streamline-diffusion finite element method with piecewise bilinear trial functions on a Shishkin mesh of O ( N 2 ) points and show that the error in the discrete space between the computed solution and the interpolant of the true solution is, uniformly in the diffusion parameter ɛ , of order ɛ 1/2 N –1 ln N + N – 3/2 in the usual streamline-diffusion norm. This includes an L 2 -norm error estimate of order O ( N – 3/2 ) in the convection–dominated case ɛ ⩽ N – 1 ln –2 N . As a corollary we prove that the method is convergent of order N –1/2 ln 3/2 N (again uniformly in ɛ ) in the local L ∞ norm on the fine part of the mesh (i.e., inside the boundary layers). This local L ∞ estimate within the layers can be improved to order ɛ 1/2 N –1/2 ln 3/2 N + N –1 ln 1/2 N , uniformly in ɛ , away from the corner layer.