The finite element methods are important and powerful tools for solving partial differential equations on complex geometries, but their high-order generalizations pose significant challenges in terms of robustness and mesh generation. In this paper, we introduce a high-order finite element method, which is insensitive to element quality and hence is more robust, and which requires only linear elements even for curved geometries and hence significantly simplifies meshing. Based on our recent work on Adaptive Extended Stencil FEM, or AES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002 nme.5246), the proposed method replaces the traditional interpolatory FEM basis functions with generalized Lagrange polynomial basis functions, constructed via local weighted least-squares approximations. In this work, we show that AES-FEM can achieve high-order accuracy using linear elements to discretize the geometry and using the standard FEM hat functions as test functions even for curved geometries. This is in contrast to most high-order generalizations of FEM, which require high-order curved elements for curved geometries. We describe the selection of stencils and the treatment of boundary conditions for high-order AES-FEM, and show that even-degree polynomial basis functions are preferred for even-order PDEs. We present numerical results in 2D and 3D for the Poisson equation and the convection-diffusion equation and demonstrate up to sixth order accuracy with AES-FEM. Overall, AES-FEM is much easier to use than FEM for curved boundaries, while being more accurate, more stable, and more efficient in terms of runtime versus error over relatively fine meshes.
Consider a partial differential equation on a bounded, simply-connected domain @math , subject to the Dirichlet and Neumann boundary conditions @math denotes a linear differential operator, @math and @math are disjoint sets of the boundary with @math , and @math denotes the outward normal to @math . A finite element method, or more generally a weighted-residual method @cite_38 , introduces a set of test functions (a.k.a. weight functions) @math , and require the residual @math to be orthogonal to @math , i.e., Eq. ) and ) are the and of the PDE, respectively. In finite element methods, @math is typically (weakly) differentiate and has a local support. In addition, @math typically forms a partition of unity, i.e., @math , so that the weighted-residual formulation is global conservative in that @math
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In this paper, we propose a method for the approximation of the solution of high- dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation for- mats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be de- signed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any ap rioriinformation on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy correc- tions in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces. Mathematics Subject Classification. 15A69, 35J50, 41A63, 65D15, 65N12.