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1. Sum factorization for fast integration of DPG matrices on prismatic elements

   https://doi.org/10.1016/j.finel.2020.103385  

   Badger, J; Henneking, S; Demkowicz, L (May 2020, Finite elements in analysis and design)

   null (Ed.)

   Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensorproduct structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order (p9) to (p7) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same (p7) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected (p7) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements. 

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2. Direct serendipity and mixed finite elements on convex quadrilaterals

   https://doi.org/10.1007/s00211-022-01274-3  

   Arbogast, Todd; Tao, Zhen; Wang, Chuning (April 2022, Numerische Mathematik)

   Abstract The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families of direct serendipity and direct mixed finite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degree r , respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for global $$H^1$$ H 1 or $$H(\text {div})$$ H ( div ) conformity, exactly two supplemental shape functions must be added to each element when $$r\ge 2$$ r ≥ 2 , and only one when $$r=1$$ r = 1 . The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index $$r\ge 1$$ r ≥ 1 , the new families of serendipity spaces $${\mathscr {DS}}_{r+1}$$ DS r + 1 are the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or full $$H(\text {div})$$ H ( div ) approximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. 10.1137/15M1013705 ). Other fully direct serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn to fully direct mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties. 

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3. The discontinuous Petrov–Galerkin method

   https://doi.org/10.1017/S0962492924000102  

   Demkowicz, Leszek; Gopalakrishnan, Jay (July 2025, Acta Numerica)

   The discontinuous Petrov–Galerkin (DPG) method is a Petrov–Galerkin finite element method with test functions designed for obtaining stability. These test functions are computable locally, element by element, and are motivated by optimal test functions which attain the supremum in an inf-sup condition. A profound consequence of the use of nearly optimal test functions is that the DPG method can inherit the stability of the (undiscretized) variational formulation, be it coercive or not. This paper combines a presentation of the fundamentals of the DPG ideas with a review of the ongoing research on theory and applications of the DPG methodology. The scope of the presented theory is restricted to linear problems on Hilbert spaces, but pointers to extensions are provided. Multiple viewpoints to the basic theory are provided. They show that the DPG method is equivalent to a method which minimizes a residual in a dual norm, as well as to a mixed method where one solution component is an approximate error representation function. Being a residual minimization method, the DPG method yields Hermitian positive definite stiffness matrix systems even for non-self-adjoint boundary value problems. Having a built-in error representation, the method has the out-of-the-box feature that it can immediately be used in automatic adaptive algorithms. Contrary to standard Galerkin methods, which are uninformed about test and trial norms, the DPG method must be equipped with a concrete test norm which enters the computations. Of particular interest are variational formulations in which one can tailor the norm to obtain robust stability. Key techniques to rigorously prove convergence of DPG schemes, including construction of Fortin operators, which in the DPG case can be done element by element, are discussed in detail. Pointers to open frontiers are presented. 

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4. Distributional Finite Element curl div Complexes and Application to Quad Curl Problems

   https://doi.org/10.1137/23M1617400  

   Chen, Long; Huang, Xuehai; Zhang, Chao (June 2025, SIAM Journal on Numerical Analysis)

   This paper addresses the challenge of constructing finite element curl div complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element curl div complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods. 

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5. Finite elements for divdiv conforming symmetric tensors in three dimensions

   https://doi.org/10.1090/mcom/3700  

   Chen, Long; Huang, Xuehai (December 2021, Mathematics of Computation)

   Finite element spaces on a tetrahedron are constructed for div div -conforming symmetric tensors in three dimensions. The key tools of the con- struction are the decomposition of polynomial tensor spaces and the charac- terization of the trace operators. First, the div div Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial com- plexes. Second, traces for the divdiv operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element div div complex, as well as the bubble functions complex, in three dimensions is established. 

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