The dynamics of beams that undergo large displacements is analyzed in frequency domain and comparisons between models derived by isogeometric analysis and p-FEM are presented. The equation of motion is derived by the principle of virtual work, assuming Timoshenko’s theory for bending and geometrical type of nonlinearity. As a result, a nonlinear system of second order ordinary differential equa...

In this paper we present a goal-oriented self-adaptive hp Finite Element Method (hp-FEM) with shared data structures and a parallel multi-frontal direct solver. The algorithm automatically generates (without any user interaction) a sequence of meshes delivering exponential convergence of a prescribed quantity of interest with respect to the number of degrees of freedom. The sequence of meshes i...

In this paper we present a new parallel algorithm of the self-adaptive hp Finite Element Method (hp-FEM) with shared data structures. The algorithm generates in a fully automatic mode (without any user interaction) a sequence of meshes delivering exponential convergence of the prescribed quantity of interest with respect to the mesh size (number of degrees of freedom). The sequence of meshes is...

We consider a finite element method (FEM) with arbitrary polynomial degree for nonlinear monotone elliptic problems. Using a linear elliptic projection, we first give a new short proof of the optimal convergence rate of the FEM in the L norm. We then derive optimal a priori error estimates in the H and L norm for a FEM with variational crimes due to numerical integration. As an application we d...

In this paper we present a parallel algorithm of the goal-oriented self-adaptive hp Finite Element Method (hp-FEM) with shared data structures and with parallel multi-frontal direct solver. The algorithm generates in a fully automatic mode (without any user interaction) a sequence of meshes delivering exponential convergence of the prescribed quantity of interest with respect to the mesh size (...

Hybrid FEM/MoM methods combine the finite element method (FEM) and the method of moments (MoM) to model inhomogeneous unbounded problems. These two methods are coupled by enforcing the continuity of tangential fields on the boundary that separates the FEM and MoM regions. When modeling complex geometries with many elements on the boundary, the MoM part of the problem is the bottleneck of the hy...

The band structure of 2D photonic crystals and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using p-FEM for smooth and hp-FEM for polygonal interfaces and boundaries. In this article we discre...

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form A(x) : D2u(x) = f(x). The fine scale is given by the meshsize h whereas the coarse scale is dictated by an integrodifferential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) for any uniformly positive definite matrix A(x) provided that the mesh is weak...

In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz boundary value problems that is based on decomposing the solution into in two parts: the first part has t...