Abstract
In this work, we implement the mortar spectral element method for the biharmonic problem with a homogeneous boundary condition. We consider a polygonal domain with corners which relies on the mortar decomposition domain technique. We propose the Strang and Fix algorithm, which permits to enlarge the discrete space of the solution by the first singular function. The interest of this algorithm is the approximation of the solution and the leading singular coefficient which has a physical significance in the propagation of cracks. We give some numerical results which confirm the optimality of the order of the error.
Highlights
IntroductionSince the discrete solution (polynomial) is regular on each sub-domain of the decomposition, the non-conformity results in the imposition of an integral matching condition on the solution and its normal derivative
Consider the fourth order problem with homogeneous boundary conditions called the biharmonic homogeneous problem⎧ ⎪⎪⎨ 2φ = f in,⎪⎪⎩φ∂∂φn==00 on ∂, on ∂, (1)where is a polygonal domain of Rd, d = 2, 3 and ∂ is a Lipschitz-continuous boundary of [1].This type of problem is involved in many problems in the mechanics of a continuous medium for both fluids and solids
To weaken the effect of the geometric singularity, we apply the method of domain decomposition without overlapping
Summary
Since the discrete solution (polynomial) is regular on each sub-domain of the decomposition, the non-conformity results in the imposition of an integral matching condition on the solution and its normal derivative. In this work we will use these results to implement the mortar spectral elements method for the Strang and Fix algorithm [9] in the case of the biharmonic problem It consists in enlarging the discrete space by the first singular function. We verify that these polynomials are represented by the formula hi(x) ci (x (1 – x2)2LN (x) ξ1)(x – ξi)(x – ξN It follows that for φδ∗ = φδ + λ1δS1 in the space Xδ∗, where λ1δ is the approximate value of the leading singular coefficient λ1, NN φδ∗(x, y)/ k =. ⎟⎟⎟⎠, where φδ∗ is the vector of admissible unknowns and φδ∗ is the vector of degrees of freedom
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