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68 Finite element methods are widely used to solve partial differential equations (PDE) in science and…
71 …asis functions is increased, resulting in $O(p^d)$ storage and $O(p^{2d})$ compute per degree of f…
76 …of implementations optimized for CPUs and GPUs, including support for just-in-time (JIT) compilati…
77 It is designed for convenient use in new and legacy software, and offers interfaces in C99 [@C99-la…
78 Users and library developers can integrate `libCEED` at a low level into existing applications in p…
79 Alternatively, users can utilize integrated `libCEED` support in MFEM [@MFEMlibrary; @mfem-paper].
81 In addition to supporting applications and discretization libraries, `libCEED` provides a platform …
89 … stabilization of the problem [@Brown:2010] and the functions $u$ and $v$ live in a suitable space.
90 Integrals in the weak form are evaluated by summing over elements $e$,
95 …idual in operator notation (\autoref{fig:decomposition}), where $\mathcal P$ is an optional extern…
96 Inhomogeneous Neumann, Robin, and nonlinear boundary conditions can be added in a similar fashion b…
101 `libCEED`'s native C interface is object-oriented, providing data types for each logical object in …
111 A `CeedOperator` is composed of one or more operators defined as in \autoref{fig:decomposition}, an…
118 …ement resolution, topology, or basis degree (see \autoref{fig:schematic}), in contrast to systems …
120 Additionally, a single source implementation (in vanilla C or C++) for the `CeedQFunction`s can be …
122 … as well as direct assembly of sparse matrices (e.g., for coarse operators in multigrid) and const…
123 …domain solvers based on the fast diagonalization method [@lottes2005hms] are in active development.
127 (in this case representing a Laplacian) on element resolution, topology, and basis degree.\label{fi…
131 `libCEED` provides high-level interfaces in Python, Julia, and Rust, each of which is maintained an…
133 …in a nearly 1:1 correspondence. For instance, a `CeedVector` object is exposed as `libceed.Vector`…
135 …in-time compilation capabilities to enable concise definition of Q-functions that work on both CPU…
141 \autoref{fig:libCEEDBackends} shows a subset of the backend implementations (backends) available in…
151 … of elements. The CPU enables faster solution of smaller problem sizes (as in strong scaling) whil…
156 … or IMEX time integration; \autoref{fig:NSvortices} shows vortices arising in the "density current…
161 ![Strain energy density in a twisted Neo-Hookean beam.\label{fig:Solids}](img/SolidTwistExample.jpe…
166 The `libCEED` implementations, accessed in this way, currently provide MFEM users with the fastest …
170 …ations, hardware, advanced system engineering and early testbed platforms, in support of the natio…