MATH 676. Finite element methods in scientific computing
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1 MATH 676 Finite element methods in scientific computing Wolfgang Bangerth, Texas A&M University
2 Lecture 21: Linear solvers for problems with more than one solution variable
3 Linear systems for vector-valued problems Recall the mixed form of the Laplace equation: The associated weak form is K 1 u+ p = 0 u = f (ϕ u, K 1 u) ( ϕ u, p) (ϕ p, u) = (ϕ p, f ) Expanding the solution and testing with discrete test functions yields this matrix: A ij = (ϕ i,u, K 1 ϕ j, u ) ( ϕ i,u,ϕ j, p ) (ϕ i, p, ϕ j,u )
4 Linear systems for vector-valued problems Let us look at this more closely: We build the bilinear form like this in code because it's convenient However, in reality, shape functions are nonzero only in either the u or p components! I.e., they are either A ij = (ϕ i,u, K 1 ϕ j, u ) ( ϕ i,u,ϕ j, p ) (ϕ i, p, ϕ j,u ) Φ i = ( ϕ i,u 0 ) or Φ i = ( 0 ϕi, p ) Note: See step-8/the vector-valued module on this.
5 Linear systems for vector-valued problems Let us look at this more closely: A ij = (ϕ i,u, K 1 ϕ j, u ) ( ϕ i,u,ϕ j, p ) (ϕ i, p, ϕ j,u ) If index i is so that the shape function is of u-type, then we get a matrix row that is in fact A ij = (ϕ i,u, K 1 ϕ j, u ) ( ϕ i,u,ϕ j, p ) If index i is of p-type, then we get A ij = (ϕ i, p, ϕ j, u ) We can do the same for index j.
6 Linear systems for vector-valued problems Let us look at this more closely: If we enumerate degrees of freedom so that all the u-types come first and then the p-types, then this leads to a block matrix Here, we get: A ij = (ϕ i,u, K 1 ϕ j, u ) ( ϕ i,u,ϕ j, p ) (ϕ i, p, ϕ j,u ) A = ( M B ) B T 0 We should use this structure in our linear solvers!
7 Linear systems for vector-valued problems Block structures in deal.ii: Block structured matrices like A = ( M B ) B T 0 are represented by classes BlockSparsityPattern/ BlockSparseMatrix Block structured vectors like U = ( u p) are represented by class BlockVector.
8 Solvers Let us consider the linear system: ( M B 0)( u p) B = ( G) F T This can equivalently be written as Mu+Bp = F B T u = G We can eliminate this variable by variable: u+ M 1 Bp = M 1 F B T M 1 B p = G+B T M 1 F
9 Solvers Let us consider the linear system: ( M B 0)( u p) B = ( G) F T We have transformed this into a decoupled system: u+ M 1 Bp = M 1 F B T M 1 B p = G+ M 1 F We call S=BT M -1 B the Schur complement of the matrix S is a symmetric, positive definite matrix We can solve the equation for p with CG
10 Solvers In order to solve the solve the system u+ M 1 Bp = M 1 F B T M 1 B p = G+ M 1 F we first need to solve with S=B T M -1 B and then with M. Problem: S is not known element by element We need to represent it in code We need to access individual blocks of A Note: The deal.ii solvers do not need to know matrix elements. They only need operator actions!
11 step-20 Representing solvers for S p = G+ M 1 F M u = F Bp in deal.ii requires us to represent S,M and present it to the CG solver. Note: We also need a preconditioner. See the tutorial program for more information!
12 Solvers in deal.ii In deal.ii, solvers are templated on both the vector and operator class: template <class Vector> class SolverCG { public: template <class Matrix, class Preconditioner> void solve (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &precondition); }; Requirements: (i) Operator and vector need to be compatible; (ii) operator needs to have a function vmult (Vector &out, const Vector &in);
13 Example Here: Schur complement operator for mixed Laplace: class SchurComplement { public: SchurComplement (const BlockSparseMatrix &A); }; void vmult (Vector<double> &dst, const Vector<double> &src) const { Vector<double> tmp1(src.size()), tmp2(src.size()); A.block(0,1).vmult (tmp1, src); some_inverse_of(a.block(0,0)) >vmult (tmp2, tmp1); A.block(1,0).vmult (dst, tmp2); }
14 Example Also necessary: Need to represent the inverse of M Need to come up with a way to precondition the Schur complement Let's just take a look at the code!
15 MATH 676 Finite element methods in scientific computing Wolfgang Bangerth, Texas A&M University
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