Our aims

• Provide basic theory of how to construct and analyse finite element methods for parabolic problems in evolving domains
• This includes problems in evolving bulk domains, on evolving surfaces and coupled bulk-surface problems
• How to define and improve quality of moving meshes
• How to generalise (piecewise linear/isoparametric) surface finite elements to higher order elements

Model problem

Given “smoothly evolving smooth surface” find a scalar field such that

(or general parabolic form on evolving surface, evolving domain, arbitrary parameterisations, …)

Compatibility

Let be a family of separable Hilbert spaces, homeomorphisms, the pair are called compatible if

Consequences of compatibility

For a compatible pair , we can define the Hilbert space : and strong material derivative

Abstract formulation of the pde

Given an evolving Hilbert triple of compatible spaces , find such that

Well posedness

Theorem. Under appropriate assumptions on the spaces, push-forward maps and bilinear forms, the continuous problem has a unique solution with which satisfies the stability bounds

Surface finite element reference map

Let be a reference finite element (i.e. standard) with . Let satisfy:

•    •    • is injective,

and that can be decomposed into an affine and smooth part: such that has full column rank, and In this case, we call a surface finite element reference map.

Surface finite element

Let be a reference finite element (i.e. standard) with , a surface finite element reference map, and the triple given by

• (the element domain)
• (the shape functions)
• ( the nodal variables).

Then we call a surface finite element.

-surface finite element

If in addition

• for
• contains for all

then we call a -surface finite element.

see also (Bernardi 1989)

Examples

• reference element in (left)
• affine surface finite elements (center) (Dziuk 1988)
• isoparametric surface finite elements (right) (Heine 2005)

Norm scaling properties

If then and If then and

Interpolation property

If the reference element satisfies a Bramble-Hilbert Lemma then for all

How to bring several elements together?

We restrict to Lagrange finite elements:

Let be discrete surface with conforming subdivision . We assume that for any two adjacent (i.e. which share a common facet) surface finite elements and that

Surface finite element space

Let each have an associated surface finite element . A surface finite element space is a (proper) subset of the product space given by

Lemma. We can identify each with a continuous function.

Examples

Evolving surface finite element

We say the family is a evolving surface finite element if

• each share a common reference element

We say the family is a -evolving surface finite element if

• each is a -surface finite element

Element flow map

There exists a family of maps called the flow map given by

The flow map defines the element velocity field by

Element push forward map

The family of element push forward maps is defined for by where

Compatibility

We say that is temporally quasi-uniform if there exists such that

Lemma. If an -evolving surface finite element is temporally quasi-uniform then (for , ) and form compatible pairs.

Evolving surface finite element space

We restrict that element flow maps coincide at Lagrange points: for all we have

An evolving surface finite element space is a time-dependent family of surface finite element spaces consisting of evolving surface finite elements.

Compatibility

We define a global push forward map for by by

Lemma. If is a uniformly quasi-uniform subdivision then each element is temporally quasi-uniform. In particular (for , ) and are compatible pairs.

Material derivatives

Let . We have that form a compatible pair

• we can define the spaces and
• we can define a discrete material derivative

Lemma. Denote by the set of global basis functions. Then .

Relation to continuous problem

We don’t have time in this talk to go into details….

Relating these definitions to their continuous counterparts requires lifting operators:

• .

This also provides an interpolation operator .

Construction of initial discrete surface

At initial time use interpolation of normal projection operator to define initial surface finite element reference map. Examples shown for isoparametric elements .

Construction of evolving discrete surface

Construct discrete flow map as interpolation of smooth flow map: Lagrange points move with velocity .

Evolving surface finite element space

Let be an isoparametric finite element space over (Demlow 2009; Kovács 2018). Assume that the evolving subdivision is uniformly quasi-uniform.

Proposition. The above construction defines to be an evolving surface finite element space consisting of -evolving surface finite elements and the corresponding pair form a compatible pair.

Further analysis of finite element scheme

• We also give details of stability and error analyses based on an abstract formulation.
• The results give show quasi-optimal error bounds for a general parabolic problem on an evolving open domain, evolving surface and a coupled bulk surface problem.
• I do not have time to give any further details…

Numerical results

Consider the unit sphere and define by the velocity: For solution we take and consider a general parabolic operator with , ,

Choose , time step to recover optimal scaling.

Numerical results

Summary

• We have shown fundamental definitions of an evolving surface finite element space
• We have (briefly) considered one particular realisation using isoparametric surface finite elements
• We have given ideas of how one might generalise this theory
• We have not shown how this relates to an abstract finite element analysis to show stability and error bounds
• We have not shown similar constructions for problems in evolving bulk domains