The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples. Classification (2000) 47H05,49M15,65N12,65N15,65N30 Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis' under the project 'Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics' (CA Mathematics Subject

Abstract. This paper provides a discrete Poincaré inequality in n space dimensions on a simplex K with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on K and all integrals of jumps zero along all interior sides by its Lebesgue norm by C (n) diam(K ). The explicit constant C (n) depends only on the dimension n = 2, 3 in case of an adaptive triangulation with the newest vertex bisection. The second part of this paper proves the stability of an enrichment operator, which leads to the stability and approximation of a (discrete) quasi-interpolator applied in the proofs of the discrete Friedrichs inequality and discrete reliability estimate with explicit bounds on the constants in terms of the minimal angle ω 0 in the triangulation. The analysis allows the bound of two constants Λ 1 and Λ 3 in the axioms of adaptivity for the practical choice of the bulk parameter with guaranteed optimal convergence rates.

No abstract

No abstract

This paper presents an algorithm for the refinement of two-or threedimensional meshes with respect to an implicitly given domain, so that its surface is approximated by facets of the resulting polytopes. Using a Cartesian grid, the proposed algorithm may be used as a mesh generator. Initial meshes may consist of polytopes such as quadrilaterals and triangles, as well as hexahedrons, pyramids, and tetrahedrons. Given the ability to compute edge intersections with the surface of an implicitly given domain, the proposed marching volume polytopes algorithm uses predefined refinement patterns applied to individual polytopes depending on the intersection pattern of their edges. The refinement patterns take advantage of rotational symmetry. Since these patterns are applied independently to individual polytopes, the resulting mesh may encompass the so-called orientation problem, where two adjacent polytopes are rotated against one another. To allow for a repeated application of the marching volume polytopes algorithm, the proposed data structures and algorithms account for this ambiguity. A simple example illustrates the advantage of the repeated application of the proposed algorithm to approximate domains with sharp corners. Furthermore, finite element simulations for two challenging real-world problems, which require highly accurate approximations of the considered domains, demonstrate its applicability. For these simulations, a variant of the fictitious domain method is used. KEYWORDSfictitious domain method, finite element method, marching cubes, marching tetrahedrons, mesh generation, volume mesh Delaunay, quadtree/octree, and advancing-front methods for mesh generation. Using quadtree or octree techniques, 2-6 Cartesian grids covering a given domain are repeatedly refined via isotropic bisections of quadrilaterals into four similar sub-quadrilaterals or of hexahedrons into eight similar sub-hexahedrons for