By Murat Uzunca

**Read Online or Download Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows (Lecture Notes in Geosystems Mathematics and Computing) PDF**

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**Additional info for Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows (Lecture Notes in Geosystems Mathematics and Computing)**

**Example text**

There are mainly two variants of the GLSFEMs; the stabilized and the direct versions. 34). 36) K where the stabilization parameter is deﬁned on each element K as [52] τK = 1 4ε h2 + 2|β | h + |α| . : for all vh ∈ Vh , ﬁnd uh ∈ Uh such that a(uh , vh ) + (r(uh ), vh )L2 (Ω ) + ∑ τK J˜K (uh , vh ) = ( f , vh )L2 (Ω ) . 4 Comparison with Galerkin Least Squares FEM (GLSFEM) 45 Note that the stabilized continuous FEM,streamline upwind Petrov-Galerkin (SUPG) method is obtained by setting SK (uh , vh ) = (L uh + r(uh ) − f , β · ∇vh )L2 (K) with different choices of the parameter τK .

For comparison, we provide results by using BiCGStab with two block preconditioners. The preconditioning matrices M1 and M2 for the permuted 50 3 Elliptic Problems with Adaptivity full systems are given as M1 = A 0 CT S M2 = , AB . 1. Our proposed method where we compute the block LU factorization of the partitioned matrix and solve the system involving the Schur complement iteratively via preconditioned BiCGStab is the best in terms of the total time compared to other methods for both uniform and adaptive reﬁnement.

1) on Ω = (0, 1)2 , and having a Monod type non-linearity r(u) = −u/(1 + u) and homogeneous source function. The advection ﬁeld and the diffusion coefﬁcient are given as β (x1 , x2 ) = (−x2 , x1 )T and ε = 10−6 , respectively. 5 Numerical Examples 3HUPXWHGZRSUHF 3HUPXWHGZSUHF0 3HUPXWHGZSUHF0 6FKXUZRSUHF 6FKXUZSUHFLOX6 %LFJVWDELWV %LFJVWDELWV 51 3HUPXWHGZRSUHF 3HUPXWHGZSUHF0 3HUPXWHGZSUHF0 6FKXUZRSUHF 6FKXUZSUHFLOX6 'R)V 'R)V Fig.