Two-dimensional HP-adaptive Algorithm for Continuous Approximations of Material Data Using Space Projection

In this paper we utilize the concept of the L2 and H1 projections used toadaptively generate a continuous approximation of an input material data inthe finite element (FE) base. This approximation, along with a correspondingFE mesh, can be used as material data for FE solvers. We begin with a brieftheoretical background, followed by description of the hp-adaptive algorithmadopted here to improve gradually quality of the projections. We investigatealso a few distinct sample problems, apply the aforementioned algorithms andconclude with numerical results evaluation

Efficient Large Scale Electromagnetics Simulations Using Dynamically Adapted Meshes with the Discontinuous Galerkin Method

A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials (p-adaptation) as well as their combination. The computation of the approximation within locally adapted elements is based on projections between finite element spaces (FES), which are shown to preserve an upper limit of the electromagnetic energy. The formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency. Error and smoothness estimates based on interface jumps are presented and applied to the fully hp-adaptive simulation of two examples in one-dimensional space. A full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three-dimensional space.Comment: 33 pages, 8 figures, submitted to Journal of Computational and Applied Mathematic

Application of projection-based interpolation algorithm for non-stationary problem

In this paper we present a solver for non-stationary problems using L2 projection and h-adaptations. The solver utilizes the Euler time integration scheme for time evolution mixed with the projection based interpolation techniques for solving the L2 projections problem at every time step. The solver is tested on the model problem of the heat transfer in L-shape domain. We show that our solver delivers linear computational cost at every time step

A discontinuous Galerkin moving mesh method for Hamilton-Jacobi equations

In this paper we consider the numerical solution of first-order Hamilton-Jacobi equations using the combination of a discontinuous Galerkin finite element method and an adaptive $r$-refinement (mesh movement) strategy. Particular attention is given to the choice of an appropriate adaptivity criterion when the solution becomes discontinuous. Numerical examples in one and two dimensions are presented to demonstrate the effectiveness of the adaptive procedure

hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated