# FEM assemblers and solvers

Important notes

• new users should read the tutorial on solving linear PDEs before reading this; this page is more on how the assemblers and solvers work internally.
• this page only refers to the main assemblers and solvers, which use linear basis functions, ie those in the pde folder. The continuum mechanics assemblers (in the continuum_mechanics folder) use quadratic basis functions and are outside this hierarchy, although they are going towards a similar design.
• the lecture notes linked here may also be useful

# Quick Summary

Assemblers

The class AbstractFeVolumeIntegralAssembler is a generic assembler class (linear bases only) and provides the basic functionality for assembling finite element matrices (for example, a stiffness matrix, or mass matrix, or the matrix A in Ax=b), or vectors (eg b in Ax=b) based on volume integrals. Concrete classes which inherit from AbstractFeVolumeIntegralAssembler say whether they want to assemble a matrix or vector (or both); and if they are assembling a matrix, implement the pure-like method ComputeMatrixTerm() which says exactly what the matrix to be assembled should be. Similarly concrete assemblers that assemble vectors implement ComputeVectorTerm().

Solvers

The main solvers now only solve linear PDEs. There is an abstract static an abstract dynamic solver, defining the following interfaces:

AbstractStaticLinearPdeSolver has

• a Solve() method

AbstractDynamicLinearPdeSolver has

• a SetTimes() method
• a SetInitialCondition() method
• a Solve() method

Concrete classes need to implement the pure method SetupLinearSystem(), which should completely setup the linear system to be solved (each timestep). How they set it up will depend on both the PDE solved and the precise numerical scheme used. They can now make use of as many assemblers as they want.

Hybrid

The class AbstractAssemblerSolverHybrid is a useful class for allowing concrete classes of the old design: classes that are both assemblers and solvers. AbstractAssemblerSolverHybrid inherits from AbstractFeVolumeIntegralAssembler, and has a method for constructing a linear system using its parent assembler. Concrete classes should inherit from it and the static/dynamic solver (as appropriate). See below for more details.

Cardiac assemblers and solvers

See section in full description (more or less self-contained).

# Full Description

Finite element solution of PDEs involves solving a linear system, whether it is (i) the single linear system solved in the FEM discretisation of a static linear problem; or (ii) the linear system solved each timestep of a FEM discretisation of a time-dependent linear problem; or (iii) the linear system solved each newton iteration of a nonlinear problem.

The previous assembler hierachy was actually an assembler-solver hierachy, and based on the notion that every problem involved defining how to do the finite element assembly of the left-hand-side (LHS) matrix and how to do the finite element assembly of the right-hand side (RHS) vector, ie the matrix A and vector b in the linear system Ax=b. Here, finite element assembly means looping over elements, doing numerical integration on each element, and adding small matrices/vectors to A or b. This led to the solver-assembler hierachy, where the concrete solver-assemblers were partly a PDE solver interface and partly an assembler of LHS matrix A and RHS vector b.

This worked fine for simple FEM implementations. However, for more advanced implementations, such as monodomain problems with matrix-based-RHS, where the linear system is of the form Ax=Mz, where A and M (mass matrix) need to be assembled in a finite element manner, but not z, this doesn't fit in very well. An even worse example is monodomain with a 'correction' term, where the linear system is Ax=Mz+c, where A,M and c need to be assembled in a finite element manner.

This led to the new assemblers and solvers, which is no longer a complex assembler-solver hierachy, but a simple assembler hierarchy, and a separate solver hierarchy.

## AbstractFeVolumeIntegralAssembler

### Introduction

This class is used for creating any matrix or vector which needs to be assembled in a finite element manner via volume integrals, ie, the matrix A or vector b in Ax=b; or A or M or c in Ax=Mz+c. It currently uses linear basis functions only.

The class is templated over element, spatial and problem dimensions as usual, but also on two booleans, CAN_ASSEMBLE_VECTOR and CAN_ASSEMBLE_MATRIX:

template<unsigned ELEM_DIM,unsigned SPACE_DIM,unsigned PROBLEM_DIM,bool CAN_ASSEMBLE_VECTOR, bool CAN_ASSEMBLE_MATRIX, InterpolationLevel INTERPOLATION_LEVEL>
class AbstractFeVolumeIntegralAssembler


The methods which need to be implemented by the concrete classes are:

ComputeMatrixTerm();
ComputeVectorTerm();
ComputeVectorSurfaceTerm();


These methods are essentially the pure methods which need to be implemented by the concrete class. Technically, though, they are not pure: they have a default implementation (which is to throw an error), so that the concrete classes just need to implement the ones that will be needed.

For example: To create a concrete class which is to be used for assembling the mass matrix M in 1d, the user has to write a concrete class inheriting from AbstractFeVolumeIntegralAssembler<1,1,1,false,true,NORMAL>, and then implement the ComputeMatrixTerm() methods (the other two methods don't need overloading as no vectors are assembled). Similarly, to create a concrete class for assembling a vector, we inherit from AbstractFeVolumeIntegralAssembler<1,1,1,true,false,NORMAL> and implement ComputeVectorTerm(). The calling code then needs to provide a constructed Mat or Vec, and the assembler will assemble them. See pde/test/TestAbstractFeVolumeIntegralAssembler.hpp for examples of concrete classes and usage.

### ComputeMatrixTerm(), ComputeVectorTerm()

These are the methods the concrete assembler has to implement. They need to provide the integrand of the intregral being evaluated at the current quadrature point, which will be a vector/matrix of size num_nodes_in_element

For example: Suppose we are in 2d, and want to set up the vector b, where

b_i = \integral_{\Omega} f(x) \phi_i(x) dV


which is the RHS vector needed to solve linear elliptic equations with source term f.

We inherit from AbstractFeVolumeIntegralAssembler<1,1,1,true,false,NORMAL> and implement ComputeVectorTerm(). This has parameters which include the quadrature point rX, and the vector of basis functions evaluated at that quad point rPhi ( = [\phi_1(x), \phi_2(x), \phi_3(x)]):

c_vector<double,PROBLEM_DIM*(ELEMENT_DIM+1)> ComputeVectorTerm(
c_vector<double, ELEMENT_DIM+1>& rPhi,
ChastePoint<SPACE_DIM>& rX,
c_vector<double,PROBLEM_DIM>& rU,
Element<ELEMENT_DIM,SPACE_DIM>* pElement)


This implementation needs to return the vector [f(x)phi_1(x), f(x)phi_2(x), f(x) phi_3(x)]. It will therefore use rX and rPhi from the parameters, but not any of the others.

ComputeMatrixTerm() is similar.

### InterpolationLevel

In the above example the parameters rU, rGradPhi, rGradU are not needed by the concrete class, and therefore didn't need interpolating onto the current quadrature points. Some control is available over which quantities are interpolated through the final template parameter of the abstract class

template<unsigned ELEM_DIM,unsigned SPACE_DIM,unsigned PROBLEM_DIM,bool CAN_ASSEMBLE_VECTOR, bool CAN_ASSEMBLE_MATRIX, InterpolationLevel INTERPOLATION_LEVEL>
class AbstractFeVolumeIntegralAssembler


For example if InterpolationLevel is not equal to NONLINEAR, rGradPhi is not computed for vectors. See class documentation for available options. Non-interpolated parameters are just passed in which zeroed values, so choosing an incorrect InterpolationLevel will lead to the solver silently giving completely incorrect answers if a quantity is used when it hasn't been interpolated.

### AbstractFeSurfaceIntegralAssembler - surface integrals

For integrals over surface elements (ie the terms coming from Neumann boundary conditions), there is a similar abstract class AbstractFeSurfaceIntegralAssembler which the same interface. For assembling surface-integral contributions to a vector, the concrete class needs to provide a ComputeVectorSurfaceTerm(). There is a concrete NaturalNeumannSurfaceTermAssembler which is used in most problems.

## The solvers of linear PDEs

### Introduction

There are three relatively simple solver classes for linear PDEs: AbstractStaticLinearPdeSolver for solving linear time-independent problems, AbstractDynamicLinearPdeSolver for solving linear time-dependent problems, and a parent which includes some common functionality

                    AbstractLinearPdeSolver
^            ^
|            |
|            |
AbstractStaticLinearPdeSolver     AbstractDynamicLinearPdeSolver


AbstractStaticLinearPdeSolver has

• a Solve() method

AbstractDynamicLinearPdeSolver has

• a SetTimes() method (for setting start time, end time and timestep).
• a SetInitialCondition() method
• a Solve() method

The three classes are abstract because of the following pure method in AbstractLinearPdeSolver

• void SetupLinearSystem()=0;

This is the method that needs to be implemented by a concrete solver. It is called by the two Solve methods, and should completely set up the linear system to be solved (that timestep), using whichever assemblers it needs and applying boundary conditions if necessary.

### AbstractAssemblerSolverHybrid

Now, in principle, to solve a particular PDE, we need to create a concrete assembler, MyAssembler, for that PDE, and then create a concrete solver, MySolver, which implements SetupLinearSystem using MyAssembler. This would be a "has-a" solver-assembler relationship ("solver has an assembler"), and is the new design.

For convenience, the old "is-a" relationship ("solver is also an assembler") is still allowed, through the class AbstractAssemblerSolverHybrid. This can be used when an assembler will be written which will define the whole of the LHS matrix A, and the whole of the RHS vector b.

AbstractAssemblerSolverHybrid inherits from

AbstractFeVolumeIntegralAssembler<..,true,true,..>


and has a method SetupGivenLinearSystem(..,..,LinearSystem*) which sets up the given linear system using the parent-assembler part of the class.

Concrete classes should inherit from this and one of AbstractStaticLinearPdeSolver or AbstractDynamicLinearPdeSolver. They would then have the following pure methods

• ComputeMatrixTerm()
• ComputeVectorTerm()
• SetupLinearSystem (inherited from the parent solver)

The concrete class then has to implement the Compute*Term methods as appropriate for its particular PDE. For SetupLinearSystem it simply has to have a one-line function which just calls SetupGivenLinearSystem. See SimpleLinearEllipticSolver as an example.

## Concrete solvers

### Non-heart

• SimpleLinearEllipticSolver: inherits from AbstractAssemblerSolverHybrid and AbstractStaticLinearPdeSolver and implements Compute*Term methods appropriately.
• SimpleLinearParabolicSolver: inherits from AbstractAssemblerSolverHybrid and AbstractDynamicLinearPdeSolver and implements Compute*Term methods appropriately.
• SimpleNonlinearEllipticSolver: inherits from ("is-a") AbstractFeVolumeIntegralAssembler but is outside the solver hierarchy. See class documentation.

### Cardiac assemblers and solvers

See the FE implementations document for the discretisations being set up and solved in the cardiac electro-physiology solvers. For both monodomain and bidomain the linear system can be written as Ax=Mz, where A and M require assembling (looping over elements) but z does not, or as Ax=Mz+c, where c also has to be assembled. The "is-a" solver-assembler approach therefore cannot be used and the "has-a" design is used.

Monodomain

The assembler classes

• MonodomainAssembler
• the main monodomain assembler: can construct both the matrix A, and the vector arising from surface stimuli contributions
• MassMatrixAssembler
• for assembling the mass matrix, M
• a NaturalNeumannSurfaceTermAssembler for Neumann BCs
• also there is an assembler for assembling the correction term if SVI is used (see here)

The solver is:

• MonodomainSolver
• inherits from AbstractDynamicLinearPdeSolver
• owns a MonodomainAssembler and a MassMatrixAssembler (and if needed an assembler for the SVI term)
• uses these assemblers to set up the linear system in SetupLinearSystem()

Bidomain

Bidomain is a bit more complicated, due to the possibility of a bath and extra bidomain specific functions. See the FE implementations document for the discretisations. The basic ideas here are the same as with monodomain though.

Assemblers:

• BidomainAssembler
• the main bidomain assembler: can construct both the matrix A
• BidomainMassMatrixAssembler
• A normal MassMatrixAssembler can't be used
• can deal with both bath/non-bath problems.
• BidomainWithBathAssembler
• inherits from BidomainAssembler, overloaded method to take into account the bath when computing the LHS matrix A.
• BidomainNeumannSurfaceTermAssembler, for the vector arising from surface stimuli contributions
• also there is an assembler for assembling the correction term if SVI is used (see here)

Solvers:

• AbstractBidomainSolver
• inherits from AbstractDynamicLinearPdeSolver
• contains a lot of common bidomain functionality (null-space; setting average phi=0; bath finalisation)
• BidomainSolver
• inherits from AbstractBidomainSolver
• owns a BidomainAssembler or BidomainWithBathAssembler as appropriate, and a BidomainMassMatrixAssembler, etc
• uses these assemblers to set up the linear system in SetupLinearSystem()
                                                AbstractLinearPdeSolver /* pure SetUpLinearSystem() method */
^            ^
|            |
|            |
AbstractStaticLinearPdeSolver     AbstractDynamicLinearPdeSolver /* SetTimes(), SetInitCondition(), Solve() methods */
^  /* Solve() method */      ^                 ^     ^
|                            |                 |     |
SimpleLinearEllipticSolver ________|                            |                 |     |__________________ SimpleLinearParabolicSolver
/* is a assembler, ie also */                                 |                 |                           /* is a assembler, ie also */
/* inherits from hybrid    */                                 |                 |                           /* inherits from hybrid    */
|                 |
|              AbstractBidomainSolver
|                       ^
|                       |
|                       |
MonodomainSolver               BidomainSolver   /* both implement SetUpLinearSystem() */
//uses severals assemblers     // uses several assemblers