79 research outputs found
Space Decompositions and Solvers for Discontinuous Galerkin Methods
We present a brief overview of the different domain and space decomposition
techniques that enter in developing and analyzing solvers for discontinuous
Galerkin methods. Emphasis is given to the novel and distinct features that
arise when considering DG discretizations over conforming methods. Connections
and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table
Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticity
For the planar Navier--Lam\'e equation in mixed form with symmetric stress
tensors, we prove the uniform quasi-optimal convergence of an adaptive method
based on the hybridized mixed finite element proposed in [Gong, Wu, and Xu:
Numer.~Math., 141 (2019), pp.~569--604]. The main ingredients in the analysis
consist of a discrete a posteriori upper bound and a quasi-orthogonality result
for the stress field under the mixed boundary condition. Compared with existing
adaptive methods, the proposed adaptive algorithm could be directly applied to
the traction boundary condition and be easily implemented
HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems
Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
Generalization of Mixed Multiscale Finite Element Methods with Applications
Many science and engineering problems exhibit scale disparity and high contrast. The small scale features cannot be omitted in the physical models because they can affect the macroscopic behavior of the problems. However, resolving all the scales in these problems can be prohibitively expensive. As a consequence, some types of model reduction techniques are required to design efficient solution algorithms.
For practical purpose, we are interested in mixed finite element problems as they produce solutions with certain conservative properties. Existing multiscale methods for such problems include the mixed multiscale finite element methods. We show that for complicated problems, the mixed multiscale finite element methods may not be able to produce reliable approximations. This motivates the need of enrichment for coarse spaces.
Two enrichment approaches are proposed, one is based on generalized multiscale finite element methods (GMsFEM), while the other is based on spectral element-based algebraic multigrid (ρAMGe). The former one, which is called mixed GMs- FEM, is developed for both Darcy’s flow and linear elasticity. Application of the algorithm in two-phase flow simulations are demonstrated. For linear elasticity, the algorithm is subtly modified due to the symmetry requirement of the stress tensor.
The latter enrichment approach is based on ρAMGe. The algorithm differs from GMsFEM in that both of the velocity and pressure spaces are coarsened. Due the multigrid nature of the algorithm, recursive application is available, which results in an efficient multilevel construction of the coarse spaces.
Stability, convergence analysis, and exhaustive numerical experiments are carried out to validate the proposed enrichment approaches. Our numerical results show that the proposed methods are more efficient than the conventional methods while still being able to produce reliable solution for our targeted applications such as reservoir simulation. Moreover, the robustness of the mixed GMsFEM for linear elasticity with respect to the high contrast heterogeneity in Poisson ratio is evident from our numerical experiments. Lastly, our empirical results show good speedup and approximation by the proposed multilevel coarsening method
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