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.
The isoparametric FEM in general uses the same-degree polynomial basis uniformly. The @math -adaptive FEM generalizes it by allowing the degrees of the basis functions to vary from element to element @cite_33 @cite_24 . The @math -adaptivity may be further combined with @math -adaptivity, leading to the @math -adaptive FEM or @math -FEM @cite_16 @cite_26 . In these methods, inter-element continuity is typically enforced as constraints @cite_40 . The @math -FEM can achieve high-order convergence, with error decreasing exponentially in the degree of basis functions, provided that the mesh is nearly optimal @cite_2 . Similar to isoparametric FEM, the @math -FEM also requires curved elements for curved geometries, requires good element shapes for accuracy, and is limited to only moderately high orders due to the use of equally spaced nodes.
@cite_26 @cite_33 @cite_24 @cite_40 @cite_2 @cite_16
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In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. The implementation of @math hp-adaptivity is challenging as hanging nodes, edges, and faces have to be constrained to ensure compatibility of the shape functions. For this reason, most @math hp-code frameworks restrict themselves to @math 1-irregular meshes to ease the implementational effort. This work alleviates these difficulties by introducing a new formulation for high-order mesh adaptivity that provides full local @math hp-refinement capabilities at a comparably small implementational effort. Its main idea is the extension of the @math hp- @math d-method such that it allows for high-order overlay meshes yielding a hierarchical, multi-level @math hp-formulation of the Finite Element Method. This concept enables intuitive refinement and coarsening procedures, while linear independence and compatibility of the shape functions are guaranteed by construction. The proposed method is demonstrated to achieve exponential rates of convergence--both in terms of degrees of freedom and in run-time--for problems with non-smooth solutions. Furthermore, the scheme is used alongside the Finite Cell Method to simulate the heat flow around moving objects on a non-conforming background mesh and is combined with an energy-based refinement indicator for automatic @math hp-adaptivity. We introduce a single unifying framework for a wide range of content-aware image warping tasks using a finite element method (FEM). Existing approaches commonly define error terms over vertex finite differences and can be expressed as a special case of our general FEM model. In this work, we exploit the full generality of FEMs, gaining important advantages over prior methods. These advantages include arbitrary mesh connectivity allowing for adaptive meshing and efficient large-scale solutions, a well-defined continuous problem formulation that enables clear analysis of existing warping error functions and allows us to propose improved ones, and higher order basis functions that allow for smoother warps with fewer degrees of freedom. To support per-element basis functions of varying degree and complex mesh connectivity with hanging nodes, we also introduce a novel use of discontinuous Galerkin FEM. We demonstrate the utility of our method by showing examples in video retargeting and camera stabilization applications, and compare our results with previous state of the art methods.