Error estimates of some non linear problems Vanessa Lleras Institut - PowerPoint PPT Presentation

Error estimates of some non linear problems Vanessa Lleras Institut montpelli´ erain Alexander Grothendieck UMR CNRS 5149 CEMRACS 2015 July 29, 2015

1 Error estimates for contact problem ◮ Linear elasticity without contact and without XFEM ◮ Frictionless contact problem with XFEM ◮ A priori error estimates ◮ A posteriori error estimates 2 Error estimates for nonlinear eigenvalue problem

The continuous problem Error The problem of homogeneous isotropic linear elasticity : with estimates for f ∈ ( L 2 (Ω)) 2 , let u ∈ ( H 1 (Ω)) 2 the displacement field solution of contact problem ◮ Linear elasticity  without contact and − div σ ( u )  = f in Ω , without XFEM   ◮ Frictionless σ ( u ) = A ε ( u ) on Ω , contact problem with XFEM = 0 on Γ D ,  u ◮ A priori error   estimates σ ( u ) n = 0 on Γ N . ◮ A posteriori error estimates Error estimates for nonlinear • u is the displacement field, eigenvalue problem • ε ( u ) = ( ∇ u + t ∇ u ) / 2 is the linearized strain tensor field, • σ is the stress tensor field, • A is the Hooke’s tensor, • f = ( f 1 , f 2 ) ∈ ( L 2 (Ω)) 2 represents the volume forces, • n denotes the normal unit outward vector of Ω on ∂ Ω.

Setting of the problem • The problem is to find u ∈ V = { v ∈ ( H 1 (Ω)) 2 ; v = 0 on Γ D } such Error estimates for that contact � � problem σ ( u ) : ε ( v ) dx = f . v dx , ∀ v ∈ V ◮ Linear elasticity Ω Ω without contact and without XFEM ◮ Frictionless contact problem with XFEM ◮ A priori error estimates ◮ A posteriori error estimates Error estimates for nonlinear eigenvalue problem

Setting of the problem • The problem is to find u ∈ V = { v ∈ ( H 1 (Ω)) 2 ; v = 0 on Γ D } such Error estimates for that contact � � problem σ ( u ) : ε ( v ) dx = f . v dx , ∀ v ∈ V ◮ Linear elasticity Ω Ω without contact and without XFEM • We approximate the continuous problem by a finite element method defined ◮ Frictionless on a regular family of triangulations of the domain. contact problem with XFEM ◮ A priori error Let T h be the partition of ¯ Ω into elements. estimates ◮ A posteriori error The family of triangulations T h , h > 0 of Ω which satisfies the following estimates Error conditions : estimates for • any 2 triangles in T h share at most a common edge or a common nonlinear eigenvalue vertex (in 2D), problem • the minimal angle of all triangles in the whole family T h is bounded away from zero. is called regular

Setting of the problem • The problem is to find u ∈ V = { v ∈ ( H 1 (Ω)) 2 ; v = 0 on Γ D } such Error estimates for that contact � � problem σ ( u ) : ε ( v ) dx = f . v dx , ∀ v ∈ V ◮ Linear elasticity Ω Ω without contact and without XFEM • We approximate the continuous problem by a finite element method defined ◮ Frictionless on a regular family of triangulations of the domain. contact problem with XFEM ◮ A priori error Let T h be the partition of ¯ Ω into elements. estimates ◮ A posteriori error The family of triangulations T h , h > 0 of Ω which satisfies the following estimates Error conditions : estimates for • any 2 triangles in T h share at most a common edge or a common nonlinear eigenvalue vertex (in 2D), problem • the minimal angle of all triangles in the whole family T h is bounded away from zero. is called regular • The discrete problem is to find u h ∈ V h the unique solution of � � σ ( u h ) : ε ( v h ) dx = f . v h dx , ∀ v h ∈ V h . Ω Ω • Existence and uniqueness of the problem thanks to Lax Milgram

Error estimates Error estimates for contact problem Given a norm � . � , an approximation η to an error � e � = � u − u h � is ◮ Linear elasticity without contact and called an error estimator. without XFEM ◮ Frictionless contact problem We can distinguish 2 type of errors : with XFEM ◮ A priori error estimates • a priori estimates : allow to qualify the tendency of the ◮ A posteriori error estimates approximation properties as a function of the number of degrees Error estimates for of freedom and the amount of work necessary for the nonlinear computation of the discrete solution. eigenvalue problem • a posteriori estimates : provide a precise upper bound of the actual error after a computation has been performed. The a posteriori indicators may tell you what to do to improve the accuracy.

A priori error estimators We bound the error by a constant (not fully known) times the best Error approximation given by the projection of the exact solution onto the estimates for contact discrete space : problem ◮ Linear elasticity If u lies in H s +1 (Ω), 0 ≤ s ≤ l with P l finite elements without contact and without XFEM then ◮ Frictionless contact problem with XFEM � u − u h � H 1 (Ω) ≤ ch s � u � H s +1 (Ω) ◮ A priori error estimates ◮ A posteriori error which means that the method is convergent of order s . estimates Error estimates for • give asymptotic rates of convergence as the mesh parameter h nonlinear eigenvalue tends to zero problem • give information about stability of various solvers • require regularity conditions of the solution which are in general not available (because of singularities) • based on the stability properties of the discrete operator • insufficient since they only yield information on the asymptotic behavior.

A posteriori estimates Error estimates for contact An a posteriori error estimation verifies : problem ◮ Linear elasticity ◮ A global upper bound : without contact and without XFEM ◮ Frictionless � contact problem � u − u h � 2 η T ( u h ) 2 1 , Ω ≤ C with XFEM ◮ A priori error estimates T ∈ T h ◮ A posteriori error estimates ◮ A local lower bound Error estimates for � nonlinear η T ( u h ) 2 ≤ C T � u − u h � 2 eigenvalue T ′ problem T’ near T ◮ Asymptotic exactness : � T ∈ T h η T ( u h ) 2 tends to 1 when the mesh size converges to � u − u h � 2 1 , Ω zero

A posteriori error estimators The idea of a posteriori estimation is to determine the order of the Error error without knowing the exact solution of the problem. estimates for contact problem • can be extracted from the numerical solution and the given data ◮ Linear elasticity without contact and of the problem which make them computable without XFEM ◮ Frictionless contact problem • are less expensive to calculate than the computation of the with XFEM ◮ A priori error numerical solution estimates ◮ A posteriori error estimates • are based on the stability properties of the continuous operator Error estimates for • have global upper bounds which are sufficient to obtain a nonlinear eigenvalue numerical solution with the accuracy below a prescribed problem tolerance. • employ information about the continuous problem. • can evaluate the quality of the finite element computations by locating the zones where the error is important and we can couple these informations with a mesh adaptivity technique which provides the user with the desired quality and minimizes the computation costs

Residual techniques Error estimates for The residual technique was introduced by Babuska and contact problem Rheinboldt in 1978. ◮ Linear elasticity without contact and without XFEM The residual is defined by : ◮ Frictionless contact problem with XFEM ◮ A priori error ( R h , v ) = a ( u h , v ) − ( f , v ) ∀ v ∈ V estimates ◮ A posteriori error estimates Error We have estimates for ( R h , v ) = a ( u h − u , v − v h ) nonlinear eigenvalue problem Therefore the error of the approximation is determined by the residual norm : ( R h , v ) � R h � H − 1 = sup v � =0 � v � V The error estimate is based on the residual norm estimates.

Residual techniques We have Error � � estimates for contact a ( u − u h , v ) = ( f + div σ ( u h ))( v − v h ) ( problem T ◮ Linear elasticity T ∈ T h without contact and � without XFEM � � � � � − 1 ◮ Frictionless σ ( u h ) n E ( v − v h )) contact problem with XFEM 2 E ◮ A priori error E ∈ E int T ∪ E N estimates T ◮ A posteriori error estimates And a is elliptic. So Error estimates for a ( u − u h , v ) nonlinear � u − u h � H 1 ≤ c sup eigenvalue � v � H 1 problem v ∈ H 1 0 (Ω) and � a ( u − u h , v ) = ≤ ( � f + div σ ( u h ) � L 2 ( T ) � v − v h � L 2 ( T ) T ∈ T h � � � � � + 1 � E � L 2 ( E ) � v − v h � L 2 ( E ) ) σ ( u h ) n 2 E ∈ E int T ∪ E N T

Clement quasi-interpolation operator Error estimates for contact problem We choose v h = R h v ◮ Linear elasticity without contact and without XFEM ◮ Frictionless For all v ∈ H 1 0 (Ω), the operator R h has the following contact problem with XFEM ◮ A priori error approximation properties : estimates ◮ A posteriori error estimates � v − R h v � L 2 ( T ) ≤ c 1 h T � v � H 1 ( ω T ) Error estimates for nonlinear eigenvalue and problem � v − R h v � L 2 ( E ) ≤ c 2 h 1 / 2 E � v � H 1 ( ω E ) The constants c 1 and c 2 are difficult to evaluate and h T is the diameter of a triangle T, h E is the length of an edge E.