This paper presents the design of flexible interfaces between finite element (FE) codes and solvers of linear equations. The main goal of the design is to allow for coupling FE codes that use different formulations (linear, non-linear, time dependent, stationary, scalar, vector) and different approximation techniques (different element types, different approximation spaces - linear, higher order, continuous, discontinuous, *h*- and *hp*-adaptive) with solvers of linear equations that use different storage formats for sparse system matrices and different solution strategies (such as, e.g., reordering of degrees of freedom (DOFs), multigrid solution or preconditioning for iterative solvers, frontal and multi-frontal strategies for direct solvers). Suitable data structures associated with the design are presented and examples of algorithms related to the interface between the FEM codes and linear solvers, together with their execution time and performance estimates, are described.

*Keywords:* finite element method, solvers of linear equations, hp-adaptivity, multigrid, multi-frontal strategies.

Edyta Hetmaniok, Iwona Nowak, Damian Słota, Adam Zielonka. Homotopy approach for solving two-dimensional integral equations of the second kind. CAMES 2016 (23)

In this paper, the two-dimensional linear and nonlinear integral equations of the second kind is analyzed. The homotopy analysis method (HAM) is used for determining the solution of the investigated equation. In this method, a solution is sought in the series form. It is shown that if this series is convergent, its sum gives the solution of the considered equation. The sufficient condition for the convergence of the series is also presented. Additionally, the error of approximate solution, obtained as partial sum of the series, is
estimated. Application of the HAM is illustrated by examples.

*Keywords:* homotopy analysis method, convergence, error estimation, nonlinear integral equation, linear integral equation.

Jan Ja¶kowiec. The

This paper deals with *hp*-type adaptation in the discontinuous Galerkin (DG) method. The DG method is formulated in this paper with a non-zero mesh skeleton width, which leads to a version of the method called in this paper the interface discontinuous Galerkin (IDG) method. In this formulation, the mesh skeleton has a finite volume and special finite elements are used for discretization. The skeleton spatial calculations are performed using the finite difference or mid-values formulas which are based on the shape functions of the neighbouring finite elements. The Dirichlet boundary conditions are applied using a non-zero width of the material between the outer boundary and a finite element aligned with the boundary. Next, the paper discusses the mesh refinement of *hp* type. In the IDG method, the mesh does not have to be conforming. The Zienkiewicz-Zhu (ZZ) error indicator is adapted in the IDG method for the purpose of mesh refinement. The paper is illustrated with two-dimensional examples, in which the mesh refinement for an elliptic problem is performed.

*Keywords:* discontinuous Galerkin method, *hp* refinement, Zienkiewicz-Zhu error estimation.

Jan Kucwaj. The impact of the Dirichlet boundary conditions on the convergence of the discretized system of nonlinear equations for potential problems. CAMES 2016 (23)

The purpose of this paper is the analysis of numerical approaches obtained by describing the Dirichlet boundary conditions on different connected components of the computational domain boundary for potential flow, provided that the domain is a rectangle. The considered problem is a potential flow around an airfoil profile. It is shown that in the case of a rectangular computational domain with two sides perpendicular to the speed direction, the potential function is constant on the connected components of these sides. This allows to state the Dirichlet conditions on the considered parts of the boundary instead of the potential jump on the slice connecting the trail edge with the external boundary. Furthermore, the adaptive remeshing method is applied to the solution of the considered problem.

*Keywords:* adaptation, rate of convergence, remeshing, Delaunay triangulation, finite element method, potential flow, Kutta-Joukovsky condition, Dirichlet condition.

CISM Programme 2016. CAMES 2016 (23)