Construction of a pp-adaptive continuous Residual Distribution scheme

A \textit{p}-adaptive continuous Residual Distribution scheme is proposed in this paper.Under certain conditions, primarily the expression of the total residual on a given element $K$ into residuals on the sub-elements of $K$ and the use of a suitable combination of quadrature formulas,it is possible to change locally the degree of the polynomial approximation of the solution.The discrete solution can then be considered non continuous across the interface of elements of different orders, while the numerical scheme still verifies the hypothesis of the discrete Lax-Wendroff theorem which ensures its convergenceto a correct weak solution.We detail the theoretical material and the construction of our \textit{p}-adaptive method in the frame of a continuous Residual Distribution scheme. Different test cases for non-linear equations at different flow velocities demonstrate numerically the validity of the theoretical results

A posteriori modeling error estimates in the optimization of two-scale elastic composite materials

The a posteriori analysis of the discretization error and the modeling error is studied for a compliance cost functional in the context of the optimization of composite elastic materials and a two-scale linearized elasticity model. A mechanically simple, parametrized microscopic supporting structure is chosen and the parameters describing the structure are determined minimizing the compliance objective. An a posteriori error estimate is derived which includes the modeling error caused by the replacement of a nested laminate microstructure by this considerably simpler microstructure. Indeed, nested laminates are known to realize the minimal compliance and provide a benchmark for the quality of the microstructures. To estimate the local difference in the compliance functional the dual weighted residual approach is used. Different numerical experiments show that the resulting adaptive scheme leads to simple parametrized microscopic supporting structures that can compete with the optimal nested laminate construction. The derived a posteriori error indicators allow to verify that the suggested simplified microstructures achieve the optimal value of the compliance up to a few percent. Furthermore, it is shown how discretization error and modeling error can be balanced by choosing an optimal level of grid refinement. Our two scale results with a single scale microstructure can provide guidance towards the design of a producible macroscopic fine scale pattern

An adaptive GMsFEM for high-contrast flow problems

In this paper, we derive an a-posteriori error indicator for the Generalized Multiscale Finite Element Method (GMsFEM) framework. This error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high-contrast coefficients. The GMsFEM, which has recently been introduced in [12], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) the construction of the online space (the latter for parameter-dependent problems). In [12], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we study an adaptive enrichment procedure and derive an a-posteriori error indicator which gives an estimate of the local error over coarse grid regions. We consider two kinds of error indicators where one is based on the $L^2$-norm of the local residual and the other is based on the weighted $H^{-1}$-norm of the local residual where the weight is related to the coefficient of the elliptic equation. We show that the use of weighted $H^{-1}$-norm residual gives a more robust error indicator which works well for cases with high contrast media. The convergence analysis of the method is given. In our analysis, we do not consider the error due to the fine-grid discretization of local problems and only study the errors due to the enrichment. Numerical results are presented that demonstrate the robustness of the proposed error indicators.Comment: 26 page

A Direct Multigrid Poisson Solver for Oct-Tree Adaptive Meshes

We describe a finite-volume method for solving the Poisson equation on oct-tree adaptive meshes using direct solvers for individual mesh blocks. The method is a modified version of the method presented by Huang and Greengard (2000), which works with finite-difference meshes and does not allow for shared boundaries between refined patches. Our algorithm is implemented within the FLASH code framework and makes use of the PARAMESH library, permitting efficient use of parallel computers. We describe the algorithm and present test results that demonstrate its accuracy.Comment: 10 pages, 6 figures, accepted by the Astrophysical Journal; minor revisions in response to referee's comments; added char