A comparison of adaptive refinement techniques for elliptic problems
ACM Transactions on Mathematical Software (TOMS)
Adaptive refinement for arbitrary finite-element spaces with hierarchical bases
Journal of Computational and Applied Mathematics
Local modification of meshes for adaptive and/or multigrid finite-element methods
Journal of Computational and Applied Mathematics
On the shape of tetrahedra from bisection
Mathematics of Computation
An explicit finite element method for the wave equation
Applied Numerical Mathematics - Special issue: a festschrift to honor Professor Robert Vichnevetsky on his 65th birthday
Local bisection refinement for N-simplicial grids generated by reflection
SIAM Journal on Scientific Computing
Quality local refinement of tetrahedral meshes based on bisection
SIAM Journal on Scientific Computing
Computing
Error estimates for finite element methods for scalar conservation laws
SIAM Journal on Numerical Analysis
Adaptive refinement of unstructured finite-element meshes
Finite Elements in Analysis and Design
Locally Adapted Tetrahedral Meshes Using Bisection
SIAM Journal on Scientific Computing
Unified multilevel adaptive finite element methods for elliptic problems
Unified multilevel adaptive finite element methods for elliptic problems
Layer based solutions for constrained space-time meshing
Applied Numerical Mathematics - Special issue: Applied numerical computing: Grid generation and solution methods for advanced simulations
Pixel-Exact Rendering of Spacetime Finite Element Solutions
VIS '04 Proceedings of the conference on Visualization '04
Self-adaptive time integration of flux-conservative equations with sources
Journal of Computational Physics
Adaptive spacetime meshing for discontinuous Galerkin methods
Computational Geometry: Theory and Applications
A Case Study in Tightly Coupled Multi-paradigm Parallel Programming
Languages and Compilers for Parallel Computing
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We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T, our method constructs a tetrahedral mesh of the spacetime domain Ω X [0,T] in constant running time per tetrahedron in ℝ3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.