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.
To overcome mesh dependence, meshless or meshfree methods have been proposed @cite_17 @cite_27 . Like IGA, these methods avoid the use the piecewise polynomials as basis functions. Some meshless methods are Galerkin methods with non-Lagrange basis functions, such as radial basis functions or Shepherd basis functions, which require special quadrature rules and cause complexities in enforcing Dirichlet boundary conditions. Another class of meshless methods is the generalized finite difference (GFD) methods @cite_18 , which are weighted residual methods using the Dirac delta functions at the nodes as test functions. When high-degree polynomials are used as basis functions, GFD can achieve high-order accuracy. However, unlike FEM, GFD methods cannot use integration by parts to reduce the order of derivatives, and are not globally conservative because the test functions do not form a partition of unity.
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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. We propose a meshless method for the extraction of high-quality continuous isosurfaces from volumetric data represented by multiple grids, also called "multiblock" data sets. Multiblock data sets are commonplace in computational mechanics applications. Relatively little research has been performed on contouring multiblock data sets, particularly when the grids overlap one another. Our algorithm proceeds in two steps. In the first step, we determine a continuous interpolant using a set of locally defined radial basis functions (RBFs) in conjunction with a partition of unity method to blend smoothly between these functions. In the second step, we extract isosurface geometry by sampling points on Marching Cubes triangles and projecting these point samples onto the isosurface defined by our interpolant. A surface splatting algorithm is employed for visualizing the resulting point set representing the isosurface. Because of our method's generality, it inherently solves the "crack problem" in isosurface generation. Results using a set of synthetic data sets and a discussion of practical considerations are presented. The importance of our method is that it can be applied to arbitrary grid data regardless of mesh layout or orientation. Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.