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In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope (the 4-simplex) and tetrahedral prism elements. More generally, our objective is to construct high-order finite element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex in four dimensions. This paper is a natural extension of the companion paper entitled "Conforming finite element function spaces in four dimensions, part I: Foundational principles and the tesseract" by Nigam and Williams (2024). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers. [ABSTRACT FROM AUTHOR]

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed 'Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom. [ABSTRACT FROM AUTHOR]

We present a novel computational framework for shape optimization problems arising in spectral geometry. The goal in such problems is to identify domains in R d which are the global optima of certain functions of the spectrum of elliptic operators on the domains. We propose the use of a combined finite element and Bayesian optimization (FEM–BO) framework in this context, and demonstrate the key ideas on two concrete examples. We study the Pólya–Szegö conjecture on polygons, and demonstrate that our proposed framework yields the theoretically proven result for triangles and quadrilaterals, and also provides compelling numerical evidence for the case of pentagons. We next study a variant of this conjecture for the Steklov eigenvalue problem. [ABSTRACT FROM AUTHOR]

In this paper we analyze a discontinuous Galerkin finite element method for approximating solutions to transport equations with certain nonlinearities. We consider models for age-structured populations allowing for a nonlinear removal rate with non-local boundary conditions on the in-flow boundary. The method employs a stabilizing term over the interior edges allowing for convergence in a stronger than usual norm. We establish convergence rates for general higher order basis functions and provide numerical examples consistent with this result. [ABSTRACT FROM AUTHOR]