623 research outputs found
How a nonconvergent recovered Hessian works in mesh adaptation
Hessian recovery has been commonly used in mesh adaptation for obtaining the
required magnitude and direction information of the solution error.
Unfortunately, a recovered Hessian from a linear finite element approximation
is nonconvergent in general as the mesh is refined. It has been observed
numerically that adaptive meshes based on such a nonconvergent recovered
Hessian can nevertheless lead to an optimal error in the finite element
approximation. This also explains why Hessian recovery is still widely used
despite its nonconvergence. In this paper we develop an error bound for the
linear finite element solution of a general boundary value problem under a mild
assumption on the closeness of the recovered Hessian to the exact one.
Numerical results show that this closeness assumption is satisfied by the
recovered Hessian obtained with commonly used Hessian recovery methods.
Moreover, it is shown that the finite element error changes gradually with the
closeness of the recovered Hessian. This provides an explanation on how a
nonconvergent recovered Hessian works in mesh adaptation.Comment: Revised (improved proofs and a better example
Verification of Unstructured Grid Adaptation Components
Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. Laminar flow over a delta wing and turbulent flow over an ONERA M6 wing are verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and a moment. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. The wing adaptation cases in the current study document a clear improvement to existing grid adaptation procedures. The stage is set for the infusion of verified grid adaptation into production fluid flow simulations
Maximum norm a posteriori error estimate for a 2d singularly perturbed semilinear reaction-diffusion problem
A singularly perturbed semilinear reaction-diffusion equation, posed in the unit square, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio assumption is made. Numerical results are presented that support our theoretical estimat
Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms
Given a function f defined on a bidimensional bounded domain and a positive
integer N, we study the properties of the triangulation that minimizes the
distance between f and its interpolation on the associated finite element
space, over all triangulations of at most N elements. The error is studied in
the W1p norm and we consider Lagrange finite elements of arbitrary polynomial
order m-1. We establish sharp asymptotic error estimates as N tends to infinity
when the optimal anisotropic triangulation is used. A similar problem has been
studied earlier, but with the error measured in the Lp norm. The extension of
this analysis to the W1p norm is crucial in order to match more closely the
needs of numerical PDE analysis, and it is not straightforward. In particular,
the meshes which satisfy the optimal error estimate are characterized by a
metric describing the local aspect ratio of each triangle and by a geometric
constraint on their maximal angle, a second feature that does not appear for
the Lp error norm. Our analysis also provides with practical strategies for
designing meshes such that the interpolation error satisfies the optimal
estimate up to a fixed multiplicative constant. We discuss the extension of our
results to finite elements on simplicial partitions of a domain of arbitrary
dimension, and we provide with some numerical illustration in two dimensions.Comment: 37 pages, 6 figure
Exact asymptotics of the uniform error of interpolation by multilinear splines
The question of adaptive mesh generation for approximation by splines has
been studied for a number of years by various authors. The results have
numerous applications in computational and discrete geometry, computer aided
geometric design, finite element methods for numerical solutions of partial
differential equations, image processing, and mesh generation for computer
graphics, among others. In this paper we will investigate the questions
regarding adaptive approximation of C2 functions with arbitrary but fixed
throughout the domain signature by multilinear splines. In particular, we will
study the asymptotic behavior of the optimal error of the weighted uniform
approximation by interpolating and quasi-interpolating multilinear splines
Towards goal-oriented mesh adaptation for fluid-structure interaction
In order to address fluid-structure interaction, we present an a priori analysis
for an ALE compressible flow model. This analysis is the key for an anisotropic metricbased
mesh adaptation
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