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
@cite_38
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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. We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification(UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases. We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let @math be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality @math . We will develop methods that approximate any function @math that is sparse in the BOPB, that is, @math of the form @math with @math of cardinality @math . Our method has a runtime of just @math , uses only @math function evaluations on a fixed and nonadaptive grid, and not more than @math bits of memory. For @math , the runtime @math will be less than what is required to simply enumerate the elements of the basis @math ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as "sublinear-time". This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size @math . The numerical approximation of parametric partial differential equations D(u,y)=0 is a computational challenge when the dimension d of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown in [1,2] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map y↦u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis in [1,2] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for classes of parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z↦u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying semi-axes lengths of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case d=∞. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDEs that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDEs, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allow us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions. A simple way to solve a partial differential equation in a non-rectangular domain @W is to embed the domain in a rectangle B and solve the problem (more easily) in the rectangle. To apply a Fourier spectral method on the rectangle, it is necessary to extend the inhomogeneous term in the PDE, f(x,y), to B in such a way that the extended function g(x,y) is periodic and infinitely differentiable, and yet is equal to f(x,y) everywhere in the irregular domain @W. If the boundary of @W, @[email protected], is defined as the zero isoline of a function @F(x,y), then a suitable extension is g(x,y)=f(x,y)H([email protected](x,y) constant) where H is a smoothed approximation to the step function which is defined in the main text.