BDDC(operator, injectiontype)

BDDC preconditioner for finite element operators

Arguments:

• operator::Operator: finite element operator to precondition
• injectiontype::BDDCInjectionType: type of injection into subassembled space to use

Returns:

• BDDC preconditioner object

getprimalnodes(bddc)

Compute or retrieve the primal nodes for a BDDC preconditioner

Returns:

• vector of primal nodes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify primal nodes
@assert bddc.primalnodes == [1, p, p^2 - p + 1, p^2]

# output

getsubassemblednodes(bddc)

Compute or retrieve the subassembled nodes for a BDDC preconditioner

Returns:

• vector of subassembled nodes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify subassembled nodes
@assert bddc.subassemblednodes == setdiff(1:(p^2), [1, p, p^2 - p + 1, p^2])

# output

getinterfacenodes(bddc)

Compute or retrieve the interface nodes for a BDDC preconditioner

Returns:

• vector of interface nodes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# get true interface nodes
trueinterfacenodes = [1:p..., (p^2-p+1):(p^2)...]
for i = 1:(p-2)
    push!(trueinterfacenodes, i * p + 1)
    push!(trueinterfacenodes, (i + 1) * p)
end
trueinterfacenodes = sort(trueinterfacenodes)
trueinterfacenodes = setdiff(trueinterfacenodes, bddc.primalnodes)

# verify interface nodes
@assert bddc.interfacenodes == trueinterfacenodes

# output

getinteriornodes(bddc)

Compute or retrieve the interior nodes for a BDDC preconditioner

Returns:

• vector of interior nodes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# get true interface nodes
trueinterfacenodes = [1:p..., (p^2-p+1):(p^2)...]
for i = 1:(p-2)
    push!(trueinterfacenodes, i * p + 1)
    push!(trueinterfacenodes, (i + 1) * p)
end
trueinterfacenodes = sort(trueinterfacenodes)
trueinteriornodes = setdiff(bddc.subassemblednodes, trueinterfacenodes)

# verify
@assert bddc.interiornodes == trueinteriornodes

# output

getprimalmodes(bddc)

Compute or retrieve the primal modes for a BDDC preconditioner

Returns:

• vector of primal modes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify primal nodes
@assert bddc.primalmodes == [1]

# output

getsubassembledmodes(bddc)

Compute or retrieve the subassembled modes for a BDDC preconditioner

Returns:

• vector of subassembled modes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify subassembled modes
@assert bddc.subassembledmodes == setdiff(1:((p-1)^2), [1])

# output

getinterfacemodes(bddc)

Compute or retrieve the interface modes for a BDDC preconditioner

Returns:

• vector of interface modes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# get true interface modes
trueinterfacemodes = [1:(p-1)...]
for i = 1:(p-2)
    push!(trueinterfacemodes, i * (p - 1) + 1)
end
trueinterfacemodes = sort(trueinterfacemodes)
trueinterfacemodes = setdiff(trueinterfacemodes, bddc.primalmodes)

# verify interface modes
@assert bddc.interfacemodes == trueinterfacemodes

# output

getinteriormodes(bddc)

Compute or retrieve the interior modes for a BDDC preconditioner

Returns:

• vector of interior modes for BDDC preconditioner

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify interior modes
@assert bddc.interiormodes == setdiff(bddc.subassembledmodes, bddc.interfacemodes)

# output

getsubassembledinverse(bddc)

Compute or retrieve the solver for subdomain for a BDDC preconditioner

Returns:

• solver for subdomain

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify subassembled inverse
@assert bddc.subassembledinverse ≈
        Matrix(diffusion.elementmatrix[bddc.subassemblednodes, bddc.subassemblednodes])^-1

# output

getinteriorinverse(bddc)

Compute or retrieve the solver for subdomain interior for a BDDC preconditioner

Returns:

• solver for subdomain interior

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify interior inverse
@assert bddc.interiorinverse ≈
        Matrix(diffusion.elementmatrix[bddc.interiornodes, bddc.interiornodes])^-1

# output

getschur(bddc)

Compute or retrieve the Schur complement matrix for a BDDC preconditioner

Returns:

• matrix for Schur complement

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
p = 4
diffusion = GalleryOperator("diffusion", p, p, mesh);
bddc = LumpedBDDC(diffusion);

# verify Schur complement
@assert bddc.schur ≈
        diffusion.elementmatrix[bddc.primalnodes, bddc.primalnodes] -
        diffusion.elementmatrix[bddc.primalnodes, bddc.subassemblednodes] *
        Matrix(diffusion.elementmatrix[bddc.subassemblednodes, bddc.subassemblednodes])^-1 *
        diffusion.elementmatrix[bddc.subassemblednodes, bddc.primalnodes]

# output

getmixedmultiplicity(bddc)

Compute or retrieve the diagonal matrix of mixed interface node and primal mode multiplicity for the BDDC preconditioner

Returns:

• matrix of mixed multiplicity for the BDDC preconditioner

getJDT(bddc)

Compute or retrieve the matrix mapping the jump over subdomain interface modes

Returns:

• matrix mapping the jump over subdomain interfacemodes

getprimalrowmodemap(bddc)

Compute or retrieve the matrix mapping the rows of the primal BDDC matrix to the primal symbol matrix

Returns:

• matrix mapping rows of primal BDDC matrix to primal symbol matrix

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);
bddc = LumpedBDDC(diffusion);

# verify primal row mode map
@assert bddc.primalrowmodemap ≈ [1 1 1 1]

# output

getprimalcolumnmodemap(bddc)

Compute or retrieve the matrix mapping the columns of the primal BDDC matrix to the primal symbol matrix

Returns:

• matrix mapping columns of primal BDDC matrix to primal symbol matrix

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);
bddc = LumpedBDDC(diffusion);

# verify primal column mode map
@assert bddc.primalcolumnmodemap ≈ [1; 1; 1; 1]

# output

getsubassembledrowmodemap(bddc)

Compute or retrieve the matrix mapping the rows of the subassembled BDDC matrix to the subassembled symbol matrix

Returns:

• matrix mapping rows of subassembled BDDC matrix to subassembled symbol matrix

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);
bddc = LumpedBDDC(diffusion);

# verify subassembled row mode map
@assert bddc.subassembledrowmodemap ≈
        bddc.operator.rowmodemap[bddc.subassembledmodes, bddc.subassemblednodes]

# output

getsubassembledcolumnmodemap(bddc)

Compute or retrieve the matrix mapping the columns of the subassembled BDDC matrix to the subassembled symbol matrix

Returns:

• matrix mapping columns of subassembled BDDC matrix to subassembled symbol matrix

Example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryOperator("diffusion", 3, 3, mesh);
bddc = LumpedBDDC(diffusion);

# verify subassembled column mode map
@assert bddc.subassembledcolumnmodemap ≈
        bddc.operator.columnmodemap[bddc.subassemblednodes, bddc.subassembledmodes]

# output

getmixedrowmodemap(bddc)

Compute or retrieve the matrix mapping the rows of the mixed BDDC matrix to the symbol matrix

Returns:

• matrix mapping rows of mixed BDDC matrix to symbol matrix

getmixedcolumnmodemap(bddc)

Compute or retrieve the matrix mapping the columns of the mixed BDDC matrix to the symbol matrix

Returns:

• matrix mapping columns of mixed BDDC matrix to symbol matrix

getmodepermutation(bddc)

Compute or retrieve the matrix permuting multi-component modes to standard ordering

Returns:

• matrix mapping BDDC mode ordering to standard ordering

computesymbolsinjection(bddc)

Compute or retrieve the injection operator for the BDDC symbol matrix

Returns:

• matrix providing the injection operator of BDDC symbol matrix