A unified penalty-free NatDRM based on de Rham complex converts Dirichlet conditions into three coupled natural subproblems for stable high-dimensional neural PDE solving up to 6D.
Arnold, Richard S
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A new bounded cochain extension operator for differential forms on Lipschitz domains achieves global commutativity with the exterior derivative on the complement of harmonic forms and yields uniform Poincaré inequalities plus Neumann eigenvalue bounds on non-convex domains.
Establishes equivalence of DEC cochains with generalized Whitney forms to prove convergence rates for the Hodge-Laplacian in full k-form generality on well-centered meshes.
The natural decomposition method uses a source subproblem, weighted curl correction, and scalar recovery to transfer boundary data into meshfree spaces, with equivalence proven for simple domains at the continuous level.
GPU port of entropy-stable DG Euler solver with non-conservative buoyancy terms reaches nearly 70% of 64-bit peak on A100 volume kernels, delivers 10x speedup and 13x better energy efficiency versus CPU, and preserves symmetry-based flux savings.
Plebanski's chiral 2-form formulation of GR reveals additional structure in Einstein's equations and supplies new analytical and numerical tools.
citing papers explorer
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Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems
A unified penalty-free NatDRM based on de Rham complex converts Dirichlet conditions into three coupled natural subproblems for stable high-dimensional neural PDE solving up to 6D.
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Uniformly Bounded Cochain Extensions and Uniform Poincar\'e Inequalities
A new bounded cochain extension operator for differential forms on Lipschitz domains achieves global commutativity with the exterior derivative on the complement of harmonic forms and yields uniform Poincaré inequalities plus Neumann eigenvalue bounds on non-convex domains.
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A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms
Establishes equivalence of DEC cochains with generalized Whitney forms to prove convergence rates for the Hodge-Laplacian in full k-form generality on well-centered meshes.
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A Natural Decomposition Method for Essential Boundary Conditions in Noninterpolatory Meshfree Spaces
The natural decomposition method uses a source subproblem, weighted curl correction, and scalar recovery to transfer boundary data into meshfree spaces, with equivalence proven for simple domains at the continuous level.
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GPU Performance of an Entropy-Stable Discontinuous Galerkin Euler Solver with Non-Conservative Terms
GPU port of entropy-stable DG Euler solver with non-conservative buoyancy terms reaches nearly 70% of 64-bit peak on A100 volume kernels, delivers 10x speedup and 13x better energy efficiency versus CPU, and preserves symmetry-based flux savings.
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General Relativity via differential forms -- explorations in Plebanski's Formalism for GR
Plebanski's chiral 2-form formulation of GR reveals additional structure in Einstein's equations and supplies new analytical and numerical tools.