Adaptive finite element methods for parabolic problems. I.: a linear model problem
SIAM Journal on Numerical Analysis
Adaptive local overlapping grid methods for parabolic systems in two space dimensions
Journal of Computational Physics
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Adaptive techniques for time-dependent problems
Computer Methods in Applied Mechanics and Engineering - Special issue on reliability in computational mechanics
SIAM Journal on Numerical Analysis
Parallel structures and dynamic load balancing for adaptive finite element computation
Proceedings of international centre for mathematical sciences on Grid adaptation in computational PDES : theory and applications: theory and applications
Computational Differential Equations
Computational Differential Equations
Generalized Green's Functions and the Effective Domain of Influence
SIAM Journal on Scientific Computing
Local adaptive mesh refinement for shock hydrodynamics
Journal of Computational Physics
A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
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We describe and test an adaptive algorithm for evolution problems that employs a sequence of “blocks” consisting of fixed, though nonuniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, remeshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies for achieving this goal using adjoint-based a posteriori error estimates, and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.